Original: A T HEORY OF T OURNAMENT R EPRESENTATIONS
Revised: A T HEORY OF T OURNAMENT R EPRESENTATIONS

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Real world tournaments are almost always intransitive.
Revised: Real world tournaments are almost always intransitive.

Original: Recent works have noted that parametric models which assume d dimensional node representations can effectively model intransitive tournaments.
Revised: Recent works have noted that parametric models which assume d dimensional node representations can effectively model intransitive tournaments (Rajkumar & Agarwal (2016)).

Original: However, nothing is known about the structure of the class of tournaments that arise out of any ﬁxed d dimensional representations.
Revised: However, nothing is known about the structure of the class of tournaments that arise out of any ﬁxed d dimensional representations.

Original: In this work, we develop a novel theory for understanding parametric tournament representations.
Revised: In this work, we develop a novel theory for understanding parametric tournament representations.

Original: Our ﬁrst contribution is to structurally characterize the class of tournaments that arise out of d dimensional representations.
Revised: Our ﬁrst contribution is to structurally characterize the class of tournaments that arise out of d dimensional representations.

Original: We do this by showing that these tournament classes have forbidden conﬁgurations which must necessarily be union of ﬂip classes, a novel way to partition the set of all tournaments.
Revised: We do this by showing that these tournament classes have forbidden conﬁgurations which must necessarily be union of ﬂip classes, a novel way to partition the set of all tournaments.

Original: We further characterize rank 2 tournaments completely by showing that the associated forbidden ﬂip class contains just 2 tournaments.
Revised: We further characterize ranktournaments completely by showing that the associated forbidden ﬂip class contains just 2 tournaments.

Original: Speciﬁcally, we show that the rank 2 tournaments are equivalent to locally-transitive tournaments.
Revised: Speciﬁcally, we show that the rank 2 tournaments are equivalent to locally-transitive tournaments.

Original: This insight allows us to show that the minimum feedback arc set problem on this tournament class can be solved using the standard Quicksort procedure.
Revised: This insight allows us to show that the minimum feedback arc set problem on this tournament class can be solved using the standard Quicksort procedure.

Original: For a general rank d tournament class, we show that the ﬂip class associated with a coned-doubly regular tournament of size O ( √ d ) must be a forbidden conﬁguration.
Revised: For a general rank d tournament class, we show that the ﬂip class associated with a coned-doubly regular tournament of size O ( √ d ) must be a forbidden conﬁguration.

Original: To answer a dual question, using a celebrated result of Forster & Simon (2006), we show a lower bound of O ( √ n ) on the minimum dimension needed to represent all tournaments on n nodes.
Revised: To answer a dual question, using a celebrated result of Forster & Simon (2006), we show a lower bound of O ( √ n ) on the minimum dimension needed to represent all tournaments on n nodes.

Original: For any given tournament, we show a novel upper bound on the smallest representation dimension that depends on the least size of the number of unique nodes in any feedback arc set of the ﬂip class associated with a tournament.
Revised: For any given tournament, we show a novel upper bound on the smallest representation dimension that depends on the least size of the number of unique nodes in any feedback arc set of the ﬂip class associated with a tournament.

Original: We show how our results also shed light on upper bound of sign-rank of matrices.
Revised: We show how our results also shed light on upper bound of sign-rank of matrices.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: In this work, we lay the the foundations for a theory of tournament representations.
Revised: In this work, we lay the the foundations for a theory of tournament representations.

Original: A tournament is a complete directed graph and arises naturally in several applications including ranking from pairwise preferences, sports modeling, social choice, etc.
Revised: A tournament is a complete directed graph and arises naturally in several applications including ranking from pairwise preferences, sports modeling, social choice, etc.

Original: We say that a tournament T on n nodes can be represented in d dimensions if there exists a skew symmetric matrix M ∈ R n × n of rank d such that a directed edge from i to j is present in T if and only if M ij > 0 .
Revised: We say that a tournament T on n nodes can be represented in d dimensions if there exists a skew symmetric matrix M ∈ R n × n of rank d such that a directed edge from i to j is present in T if and only if M ij > 0 .

Original: Real world tournaments are almost always intransitive (Tversky (1969);
Revised: Real world tournaments are almost always intransitive (Tversky (1969);

Original: Klimenko (2015)) and it is not known what type of tournaments can be represented in how many dimensions.
Revised: Klimenko (2015)) and it is not known what type of tournaments can be represented in how many dimensions.

Original: This is important to understand because of the following reason: As a modeler of preference relations using tournaments, it is often more natural to have structural domain knowledge such as ‘The tournaments under consideration do not have long cycles’ as opposed to algebraic domain knowledge such as ‘The rank of the skew symmetric matrix associated with the tournaments of interest is at most k ’ .
Revised: This is important to understand because of the following reason: As a modeler of preference relations using tournaments, it is often more natural to have structural domain knowledge such as ‘The tournaments under consideration do not have long cycles’ as opposed to algebraic domain knowledge such as ‘The rank of the skew symmetric matrix associated with the tournaments of interest is at most k ’ .

Original: However, algorithms that learn rankings from pairwise comparison data typically need as input the algebraic quantity - the rank of the skew symmetric matrices associated with tournaments or equivalently the dimension where they are represented (Rajkumar & Agarwal (2016)).
Revised: However, algorithms that learn rankings from pairwise comparison data typically need as input the algebraic quantity - the rank of the skew symmetric matrices associated with tournaments or equivalently the dimension where they are represented (Rajkumar & Agarwal (2016)).

Original: To bridge the gap between the structural and the algebraic world, we ask and answer two fundamental questions regarding the representations of tournaments.
Revised: To bridge the gap between the structural and the algebraic world, we ask and answer two fundamental questions regarding the representations of tournaments.

Original: 1) What structurally characterizes the class of tournaments that can be represented in d dimensions? 2)
Revised: 1) What structurally characterizes the class of tournaments that can be represented in d dimensions? 2)

Original: Given a tournament T on n nodes, what is the minimum dimension d needed to represent it? .
Revised: Given a tournament T on n nodes, what is the minimum dimension d needed to represent it? .

Original: We answer the ﬁrst question by investigating the intricate structure of the rank d tournament class via the notion of forbidden conﬁgurations.
Revised: We answer the ﬁrst question by investigating the intricate structure of the rank d tournament class via the notion of forbidden conﬁgurations.

Original: Speciﬁcally, we show that the set of forbidden conﬁg-
Revised: Speciﬁcally, we show that the set of forbidden conﬁg-

Original: [Figure]
Revised: [Figure]

Original: uration for the rank d tournament class must necessarily be a union of ﬂip classes , a novel way to partition the set of all tournaments into equivalence classes.
Revised: uration for the rank d tournament class must necessarily be a union of ﬂip classes , a novel way to partition the set of all tournaments into equivalence classes.

Original: We explicitly characterize the forbidden conﬁgurations for the rank 2 tournament class and exhibit a forbidden conﬁguration for the general rank d tournament class.
Revised: We explicitly characterize the forbidden conﬁgurations for the rank 2 tournament class and exhibit a forbidden conﬁguration for the general rank d tournament class.

Original: Speciﬁcally, we show that the rank 2 tournaments are equivalent to locally transitive tournaments , a previously studied class of tournaments (Cohen et al.
Revised: Speciﬁcally, we show that the rank 2 tournaments are equivalent to locally transitive tournaments , a previously studied class of tournaments (Cohen et al.

Original: Our results throw light on the connections between transitive and locally transitive tournaments and also lets us develop a classic Quicksort based algorithm to solve that the minimum feedback arc set problem on rank 2 tournaments with O ( n 2 ) time complexity.
Revised: Our results throw light on the connections between transitive and locally transitive tournaments and also lets us develop a classic Quicksort based algorithm to solve that the minimum feedback arc set problem on rank 2 tournaments with O ( n 2 ) time complexity.

Original: Our results for the general rank d tournament class have connections to the classic long standing Hadamard conjecture and we discuss this as well.
Revised: Our results for the general rank d tournament class have connections to the classic long standing Hadamard conjecture and we discuss this as well.

Original: Figure 1 gives a glimpse of some of the main results.
Revised: Figure 1 gives a glimpse of some of the main results.

Original: We answer the second question by proving lower and upper bounds on the smallest dimension needed to represent a tournament on n nodes.
Revised: We answer the second question by proving lower and upper bounds on the smallest dimension needed to represent a tournament on n nodes.

Original: We exhibit a lower bound of O ( √ n ) using a variation of the celebrated dimension complexity result of Forster & Simon (2006) for sign matrices.
Revised: We exhibit a lower bound of Ω( √ n ) using a variation of the celebrated dimension complexity result of Forster & Simon (2006) for sign matrices.

Original: To show upper bounds, we introduce a novel parameter associated to a tournament called the Flip Feedback Node set of a Tournament .
Revised: To show upper bounds, we introduce a novel parameter associated to a tournament called the Flip Feedback Node set of a Tournament .

Original: This quantity depends on the least number of unique nodes in any feedback arc set of an associated tournament class for the tournament of interest and upper bounds linearly the representation dimension of any tournament.
Revised: This quantity depends on the least number of unique nodes in any feedback arc set of an associated tournament class for the tournament of interest and upper bounds linearly the representation dimension of any tournament.

Original: We show how our results can be used to provide upper bounds on the classic notion of sign-rank of a matrix.
Revised: We show how our results can be used to provide upper bounds on the classic notion of sign-rank of a matrix.

Original: Previously known upper bounds for sign rank depended on the VC dimension of the associated binary function class (Alon et al.
Revised: Previously known upper bounds for sign rank depended on the VC dimension of the associated binary function class (Alon et al.

Original: Our upper bounds on the other hand have a graph theoretic ﬂavour.
Revised: Our upper bounds on the other hand have a graph theoretic ﬂavour.

Original: Organization of the Paper:
Revised: Organization of the Paper:

Original: We discuss brieﬂy in Section 2 the foundational works this paper builds upon.
Revised: We discuss brieﬂy in Section 2 the foundational works this paper builds upon.

Original: We introduce necessary preliminaries in Section 3.
Revised: We introduce necessary preliminaries in Section 3.

Original: The answer to the ﬁrst question about the structural characterization of d dimensional tournament classes span Sections 4, 5 and 6.
Revised: The answer to the ﬁrst question about the structural characterization of d dimensional tournament classes span Sections 4, 5 and 6.

Original: We devote Section 7 to answer the second question about the number of dimensions needed to represent a tournament.
Revised: We devote Section 7 to answer the second question about the number of dimensions needed to represent a tournament.

Original: Section 7.1 explores connections of our results to upper bounds on the sign rank of a sign matrix.
Revised: Section 8.1 explores connections of our results to upper bounds on the sign rank of a sign matrix.

Original: Finally, we conclude in Section 8.
Revised: Finally, we conclude in Section 10.

Revised: All proofs are defered to the Appendix.

Original:  R ELATED W ORK
Revised:  R ELATED W ORK

Original: The work in this paper builds on several pieces of work across different domains.
Revised: The work in this paper builds on several pieces of work across different domains.

Original: We summarize below the most important related works under different categories:
Revised: We summarize below the most important related works under different categories:

Original: Intransitive Pairwise Preference Models: One of the main reasons to study representations of tournaments is to model pairwise preferences.
Revised: Intransitive Pairwise Preference Models: One of the main reasons to study representations of tournaments is to model pairwise preferences.

Original: Parametric pairwise preference models that can model intransitivity have gained recent interest.
Revised: Parametric pairwise preference models that can model intransitivity have gained recent interest.

Original: Rajkumar & Agarwal (2016) develop a low rank pairwise ranking model that can model intransitivity.
Revised: Rajkumar & Agarwal (2016) develop a low rank pairwise ranking model that can model intransitivity.

Original: However, their study and results were restricted to just the transitive tournaments in these classes.
Revised: However, their study and results were restricted to just the transitive tournaments in these classes.

Original: A generalization of the classical Bradley-Terry-Luce model (Bradley & Terry (1952), Luce (2012)) was studied in Causeur & Husson (2005).
Revised: A generalization of the classical Bradley-Terry-Luce model (Bradley & Terry (1952), Luce (2012)) was studied in Causeur & Husson (2005).

Original: However, no structural characterization is known.
Revised: However, no structural characterization is known.

Original: Same holds for the more recent models studied in Chen &
Revised: Same holds for the more recent models studied in Chen &

Original: Joachims (2016), Makhijani & Ugander (2019) and Bower & Balzano (2020).
Revised: Joachims (2016), Makhijani & Ugander (2019) and Bower & Balzano (2020).

Original: Flip Classes: The notion of ﬂip classes, a novel way to partition the set of all tournaments on n nodes, was ﬁrst introduced in Fisher & Ryan (1995).
Revised: Flip Classes: The notion of ﬂip classes, a novel way to partition the set of all tournaments on n nodes, was ﬁrst introduced in Fisher & Ryan (1995).

Original: The goal however was completely different and was on studying equilibrium on certain generalized rock-paper-scissors games on tournaments.
Revised: The goal however was completely different and was on studying equilibrium on certain generalized rock-paper-scissors games on tournaments.

Original: Interestingly, and perhaps surprisingly, the notion of ﬂip classes turn out to be fundamental to our study of understanding forbidden conﬁgurations of d dimensional tournament classes.
Revised: Interestingly, and perhaps surprisingly, the notion of ﬂip classes turn out to be fundamental to our study of understanding forbidden conﬁgurations of d dimensional tournament classes.

Original: Dimension Complexity and Sign Rank: Dimension complexity and sign rank of sign pattern matrices were studied in Forster (2002).
Revised: Dimension Complexity and Sign Rank: Dimension complexity and sign rank of sign pattern matrices were studied in Forster (2002).

Original: These results have found signiﬁcant applications in learning theory and lower bounds in computational complexity (Forster & Simon (2006)).
Revised: These results have found signiﬁcant applications in learning theory and lower bounds in computational complexity (Forster & Simon (2006)).

Original: More recently, Alon et al.
Revised: More recently, Alon et al.

Original: (2016) study the sign rank for function classes with ﬁxed Vapnik-Chervonenkis (VC) dimension and show upper bounds.
Revised: (2016) study the sign rank for function classes with ﬁxed Vapnik-Chervonenkis (VC) dimension and show upper bounds.

Original: Our upper bounds however depend on certain graph theoretic properties.
Revised: Our upper bounds however depend on certain graph theoretic properties.

Original:  P RELIMINARIES
Revised:  P RELIMINARIES

Original: Tournaments: A tournament T is a complete directed graph.
Revised: Tournaments: A tournament T is a complete directed graph i.e., a graph on n nodes where for every pair of nodes ( i, j ) either an edge is oriented from i to j or j to i .

Original: The number of nodes n in T will be usually clear from the context or will be explicitly speciﬁed.
Revised: The number of nodes n in T will be usually clear from the context or will be explicitly speciﬁed.

Original: For nodes i and j in T , we say i (cid:31) T j if there is a directed edge from i to j .
Revised: For nodes i and j in T , we say i (cid:31) T j if there is a directed edge from i to j .

Original: Given a node i , we deﬁne the out and in neighbours of i as
Revised: Given a node i , we deﬁne the out and in neighbours of i as

Original: T + i =
Revised: T + i =

Original: { j : i (cid:31) T j } and T − i =
Revised: { j : i (cid:31) T j } and T − i =

Original: { j : j (cid:31) T i } respectively.
Revised: { j : j (cid:31) T i } respectively.

Original: Given a set of nodes S , we denote by
Revised: Given a set of nodes S , we denote by

Original: T ( S ) the induced sub-tournament of T on the nodes in S .
Revised: T ( S ) the induced sub-tournament of T on the nodes in S .

Original: Feedback Arc Set and Pairwise Disagreement Error:
Revised: Feedback Arc Set and Pairwise Disagreement Error:

Original: Given a permutation σ on n nodes, the feedback arc set of σ w.r.t the tournament T is deﬁned as E σ ( T ) =
Revised: Given a permutation σ on n nodes, the feedback arc set of σ w.r.t the tournament T is deﬁned as E σ ( T ) =

Original: { ( i, j ) : σ ( i ) > σ ( j ) , i > T j } .
Revised: { ( i, j ) : σ ( i ) > σ ( j ) , i > T j } .

Original: The pairwise diagreement error of σ w.r.t T is given by | E σ ( T ) | ( n 2 ) .
Revised: The pairwise diagreement error of σ w.r.t T is given by | E σ ( T ) | ( n 2 ) .

Original: It is known that ﬁnding the σ that minimizes the pairwise disagreement error w.r.t a general tournament T is a NP-hard problem (Charbit et al.
Revised: It is known that ﬁnding the σ that minimizes the pairwise disagreement error w.r.t a general tournament T is a NP-hard problem (Charbit et al.

Original: Skew Symmetric Tournament Classes: 
Revised: Skew Symmetric Tournament Classes: A square matrix M ∈ R n × n is skew symmetric if M ij = − M ji ∀ i, j .

Original: In this paper, whenever we refer to a skew symmetric matrix M ∈ R n × n , we always assume M ij (cid:54) = 0 ∀ i (cid:54) =
Revised: In this paper, whenever we refer to a skew symmetric matrix M ∈ R n × n , we always assume M ij (cid:54) = 0 ∀ i (cid:54) =

Original: j and M ii = 0 ∀ i .
Revised: j and M ii = 0 ∀ i .

Original: Given such an M , we denote by T { M } the tournament on n nodes induced by M where i (cid:31) T j ⇐⇒ M ij > 0 .
Revised: Given such an M , we denote by T { M } the tournament on n nodes induced by M where i (cid:31) T j ⇐⇒ M ij > 0 .

Original: We refer to class of tournaments induced by rank d skew symmetric matrices as the rank d tournament class .
Revised: We refer to class of tournaments induced by rank d skew symmetric matrices as the rank d tournament class .

Original: Forbidden Conﬁgurations: A tournament class T is a collection of tournaments.
Revised: Forbidden Conﬁgurations: A tournament class T is a collection of tournaments.

Original: T is said to forbid a tournament T if no tournament in T has a sub-tournament that is isomorphic to T .
Revised: T is said to forbid a tournament T if no tournament in T has a sub-tournament that is isomorphic to T .

Original: We call T a forbidden conﬁguration for T if T forbids T but does not forbid any sub-tournament of T .
Revised: We call T a forbidden conﬁguration for T if T forbids T but does not forbid any sub-tournament of T .

Original: For example, the class of all acyclic/transitive tournaments has the 3 -cycle as a forbidden conﬁguration i.e, the tournament T on three nodes i, j, k where i (cid:31) T j (cid:31) T k (cid:31) T i .
Revised: For example, the class of all acyclic/transitive tournaments has the 3 -cycle as a forbidden conﬁguration i.e, the tournament T on three nodes i, j, k where i (cid:31) T j (cid:31) T k (cid:31) T i .

Original: Representations and Tournaments: The matrix A rot ∈ R d × d is deﬁned for every even d and is a block diagonal matrix which consists of d/ 2 blocks of [0 − 1; 1 0] .
Revised: Representations and Tournaments: The matrix A rot ∈ R d × d is deﬁned for every even d and is a block diagonal matrix which consists of d/ 2 blocks of [0 − 1; 1 0] .

Original: This is the canonical representation of a non-degenerate skew symmetric matrix i.e., any non-singular skew symmetric matrix can be brought to this form by a suitable basis transformation.
Revised: This is the canonical representation of a non-degenerate skew symmetric matrix i.e., any non-singular skew symmetric matrix can be brought to this form by a suitable basis transformation.

Original: Given a set of vectors in H =
Revised: Given a set of vectors in H =

Original: { h 1 , . . .
Revised: { h 1 , . . .

Original: , h n }  ∈ R d for some even d , we refer to the set H as a tournament inducing representation if h Ti A rot h j (cid:54)
Revised: h n } where each h i ∈ R d for some even d , we refer to the set H as a tournament inducing representation if h Ti A rot h j (cid:54)

Original: = 

Revised: =

Original: for all i (cid:54)
Revised: for all i (cid:54)

Original: =
Revised: =

Original: j .
Revised: j .

Original: Furthermore, we refer to the tournament induced by H as T [ H ] where a directed edge exits from node i to j if and only if h Ti A rot h j > 0 .
Revised: Furthermore, we refer to the tournament induced by H as T [ H ] where a directed edge exits from node i to j if and only if h Ti A rot h j > 0 .

Original: It is easy to verify that h T A rot h = 0 for any h ∈ R d .
Revised: It is easy to verify that h T A rot h = 0 for any h ∈ R d .

Original: Any skew symmetric matrix M ∈ R n × n of rank d can be written as M = H T A rot H for some H ∈ R d × n .
Revised: Any skew symmetric matrix M ∈ R n × n of rank d can be written as M = H T A rot H for some H ∈ R d × n .

Original: It follows that if M = H T A rot H , then T { M } = T [ H ] .
Revised: It follows that if M = H T A rot H , then T { M } = T [ H ] .

Original: Positive Spans: A set of vectors H = { h 1 , . .
Revised: Positive Spans: A set of vectors H = { h 1 , . .

Original: . , h n }  ∈ R d is said to positively span R d if for any w ∈ R d , there exists non-negative constants c 1 , . . 
Revised: h n } where each h i ∈ R d is said to positively span R d if for any w ∈ R d , there exists non-negative constants c 1 , . . .

Original: , c n ≥ 

Revised: , c n ≥

Original: such that (cid:80) i c i h i =
Revised: such that (cid:80) i c i h i =

Original: w .
Revised: w .

Original: If H positively spans R d , then there does not exist a w ∈ R d such that w T h i > 0 ∀ i .
Revised: If H positively spans R d , then there does not exist a w ∈ R d such that w T h i > 0 ∀ i .

Original: This is a easy consequence of Farka’s Lemma.
Revised: This is a easy consequence of Farkas’ Lemma.

Original: Remark on Notation: We reiterate that we use T ( · ) , T {·} and T [ · ] to mean different objects - the tournament induced by a subset of nodes, the tournament induced by a skew symmetric matrix and the tournament induced by a representation of a set of vectors respectively.
Revised: Remark on Notation: We reiterate that we use T ( · ) , T {·} and T [ · ] to mean different objects - the tournament induced by a subset of nodes, the tournament induced by a skew symmetric matrix and the tournament induced by a representation of a set of vectors respectively.

Original: These will be usually clear from the context.
Revised: These will be usually clear from the context.

Original:  F LIP C LASSES , F ORBIDDEN C ONFIGURATIONS AND P OSITIVE S PANS
Revised:  F LIP C LASSES , F ORBIDDEN C ONFIGURATIONS AND P OSITIVE S PANS

Original: The main purpose of this section is understand the space of forbidden conﬁgurations of rank d tournaments.
Revised: The main purpose of this section is understand the space of forbidden conﬁgurations of rank d tournaments.

Original: The main result of this section shows that the forbidden conﬁgurations for rank d tournament classes occur as union of certain carefully deﬁned equivalence classes of non-isomorphic tournaments.
Revised: The main result of this section shows that the forbidden conﬁgurations for rank d tournament classes occur as union of certain carefully deﬁned equivalence classes of non-isomorphic tournaments.

Original: Towards this, we deﬁne the notion of ﬂip classes which was introduced ﬁrst in (Fisher & Ryan, 1995)
Revised: Towards this, we deﬁne the notion of ﬂip classes which was introduced ﬁrst in (Fisher & Ryan, 1995)

Original: although in a different context:
Revised: although in a different context:

Original: Deﬁnition 1.
Revised: Deﬁnition 1.

Original: Given a tournament T on n nodes and a set S ⊆ [ n ] , deﬁne φ S ( T ) to be the tournament obtained from T by reversing the orientation of all edges ( i, j ) such that i ∈ S, j ∈ ¯ S .
Revised: Given a tournament T on n nodes and a set S ⊆ [ n ] , deﬁne φ S ( T ) to be the tournament obtained from T by reversing the orientation of all edges ( i, j ) such that i ∈ S, j ∈ ¯ S .

Original: In other words, φ S ( T ) is obtained from T by reversing the edges across the cut ( S, ¯ S )
Revised: In other words, φ S ( T ) is obtained from T by reversing the edges across the cut ( S, ¯ S )

Original: = { ( i, j ) : i ∈ S, j ∈ ¯ S } .
Revised: = { ( i, j ) : i ∈ S, j ∈ ¯ S } .

Original: Deﬁnition 2.
Revised: Deﬁnition 2.

Original: A class of tournaments on n nodes is called cut-equivalent if for every pair of tournaments T , T (cid:48) in the class, there exists a S ⊆
Revised: A class of tournaments on n nodes is called cut-equivalent if for every pair of tournaments T , T (cid:48) in the class, there exists a S ⊆

Original: [ n ] such that T (cid:48) is isomorphic to φ S ( T )
Revised: [ n ] such that T (cid:48) is isomorphic to φ S ( T )

Original: It is easy to show that the set of cut-equivalent tournaments form an equivalence relation over the set of all tournaments on n nodes Fisher & Ryan (1995).
Revised: It is easy to show that the set of cut-equivalent tournaments form an equivalence relation over the set of all tournaments on n nodes Fisher & Ryan (1995).

Original: The corresponding equivalence classes are called a ﬂip classes.
Revised: The corresponding equivalence classes are called a ﬂip classes.

Original: We denote by F ( T ) the ﬂip class of T i.e., the equivalence class of all cut-equivalent tournaments to T .
Revised: We denote by F ( T ) the ﬂip class of T i.e., the equivalence class of all cut-equivalent tournaments to T .

Original: In the following theorem we show the fundamental relation between ﬂip classes and forbidden conﬁgurations.
Revised: In the following theorem we show the fundamental relation between ﬂip classes and forbidden conﬁgurations.

Original: Theorem 1.
Revised: Theorem 1.

Original: Let T , a tournament on k nodes, be a forbidden conﬁguration for some rank d tournament class.
Revised: Let T , a tournament on k nodes, be a forbidden conﬁguration for some rank d tournament class.

Original: Then every tournament in the ﬂip class of T is also a forbidden conﬁguration for the rank d tournament class.
Revised: Then every tournament in the ﬂip class of T is also a forbidden conﬁguration for the rank d tournament class.

Original: Proof.

Original: Assume there is a T (cid:48) ∈ F ( T ) which is not forbidden for rank d tournament class.

Original: Thus, ∃ H (cid:48)

Original: = { h (cid:48) 1 , . . .

Original: , h (cid:48) n } ∈ R d such that T (cid:48)

Original: =

Original: T [ H (cid:48) ] .

Original: By deﬁnition, there must also exist a S ⊆

Original: [ n ] such that T (cid:48)

Original: =

Original: φ S ( T ) .

Original: Consider the representation H obtained from H (cid:48) where h i = − h (cid:48)

Original: i for all i ∈ S and h i = h (cid:48)

Original: i otherwise.

Original: It is easy to verify that T

Original: = T [ H ] which is a contradiction to the assumption that T is a forbidden conﬁguration for rank d tournaments.

Original: Corollary 1.
Revised: Corollary 1.

Original: ( Structure of Forbidden conﬁgurations )
Revised: ( Structure of Forbidden conﬁgurations )

Original: Forbidden conﬁgurations of rank d tournament classes are unions of ﬂip classes.Proof.
Revised: Forbidden conﬁgurations of rank d tournament classes are unions of ﬂip classes.

Original: This follows directly from Theorem 1.

Original: Thus, to characterize the forbidden conﬁguration of rank d tournament classes, we need to understand the ﬂip classes.
Revised: Thus, to characterize the forbidden conﬁguration of rank d tournament classes, we need to understand the ﬂip classes.

Original: We begin with the following simple but useful deﬁnition.
Revised: We begin with the following simple but useful deﬁnition.

Original: Deﬁnition 3.
Revised: Deﬁnition 3.

Original: A tournament T is called a R -cone if there exists a node i ∗ such that
Revised: A tournament T is called a R -cone if there exists a node i ∗ such that

Original: • i ∗ (cid:31) T j for all j (cid:54)=
Revised: • i ∗ (cid:31) T j for all j (cid:54)

Original:  i  (cid:0) j (cid:31) T i ∗ for all j (cid:54) = i  (cid:1) • T ( T + i ∗ ) (cid:0) T ( T − i ∗ ) (cid:1) is isomorphic to the tournament R .
Revised: = i ∗ (cid:0) j (cid:31) T i ∗ for all j (cid:54) = i ∗ (cid:1) • T ( T + i ∗ ) (cid:0) T ( T − i ∗ ) (cid:1) is isomorphic to the tournament R .

Original: R -cones are essentially the tournament R along with an additional node that either beats or loses to nodes in R .
Revised: R -cones are essentially the tournament R along with an additional node that either beats or loses to nodes in R .

Original: R -cones are useful as they be viewed in some sense as canonical tournaments in ﬂip classes.
Revised: R -cones are useful as they be viewed in some sense as canonical tournaments in ﬂip classes.

Original: This is justiﬁed because of the following observation.
Revised: This is justiﬁed because of the following observation.

Original: Proposition 1.
Revised: Proposition 1.

Original: Every ﬂip class contains an R -cone for some tournament R .
Revised: Every ﬂip class contains an R -cone for some tournament R .

Original: Proof.

Original: Consider any tournament T .

Original: Let T (cid:48) =

Original: φ { i ∪ T − i } ( T ) for an arbitrary node i .

Original: By deﬁnition

Original: T (cid:48) ∈ F ( T ) .

Original: Also, T (cid:48) is a R -cone,

Original: coned by i . The above observation says that to identify the forbidden conﬁgurations for a given tournament class, it sufﬁces to identify all forbidden R -cones.
Revised: The above observation says that to identify the forbidden conﬁgurations for a given tournament class, it sufﬁces to identify all forbidden R -cones.

Original: Then by Corollary 1, the associated ﬂip classes will be the set of all forbidden conﬁgurations.
Revised: Then by Corollary 1, the associated ﬂip classes will be the set of all forbidden conﬁgurations.

Original: However, it does not throw light on what property the tournament R must satisfy.
Revised: However, it does not throw light on what property the tournament R must satisfy.

Original: The following lemma establishes this.
Revised: The following lemma establishes this.

Original: Lemma 2.
Revised: Lemma 2.

Original: Let R be a tournament with the property that if R = R [ H ] for some representation
Revised: Let R be a tournament with the property that if R = R [ H ] for some representation

Original: H = { h 1 , . .
Revised: H = { h 1 , . .

Original: , h n } ∈ R d then H positively spans R d .
Revised: , h n } ∈ R d then H positively spans R d .

Original: Then rank d tournament class forbids
Revised: Then rank d tournament class forbids

Original: R -cones.
Revised: R -cones.

Original: Proof.

Original: Consider any representation H that realizes R .

Original: By assumption of the theorem, H positively spans R d .

Original: Thus, by Farka’s lemma, there cannot exist a v ∈ R d such that v T h i > 0 ∀ i .

Original: Note that if R -cone is not a forbidden conﬁguration for rank d tournaments, then there must exist a h ∈ R d such that h T A rot h i > 0 ∀ i .

Original: As A rot is invertible, one can set v =

Original: ( A rot ) T h with the property that v T h

Original: i > 0 ∀ i .

Original: But this contradicts the conclusion drawn earlier from Farka’s lemma.

Original: The above lemma is extremely helpful in the sense that it reduces the study of forbidden conﬁgurations to the study of ﬁnding tournaments such that any representation that realizes it must necessarily positively span the entire Euclidean space.
Revised: The above lemma is extremely helpful in the sense that it reduces the study of forbidden conﬁgurations to the study of ﬁnding tournaments such that any representation that induces it must necessarily positively span the entire Euclidean space.

Original: Note that there may be several representations which positively span the entire space.
Revised: Note that there may be several representations which positively span the entire space.

Original: This does not mean their the associated coned tournaments are forbidden conﬁgurations.
Revised: This does not mean their the associated coned tournaments are forbidden conﬁgurations.

Original: Instead, we start with an R cone and conclude it is forbidden if every representation that realizes R necessarily positively spans the entire space.
Revised: Instead, we start with an R cone and conclude it is forbidden if every representation that induces R necessarily positively spans the entire space.

Original: It is a non-trivial problem to identify such tournaments R for an arbitrary dimension d .
Revised: It is a non-trivial problem to identify such tournaments R for an arbitrary dimension d .

Original: In the following sections, we explicitly identify the only forbidden ﬂip class for rank 2 tournaments, one forbidden ﬂip class for rank 4 and then (a potentially weaker) forbidden ﬂip class for the general rank d case.
Revised: In the following sections, we explicitly identify the only forbidden ﬂip class for rank 2 tournaments, one forbidden ﬂip class for rank 4 and then (a potentially weaker) forbidden ﬂip class for the general rank d case.

Original:  R ANK 2 T OURNAMENTS ⇐⇒ L OCALLY T RANSITIVE T OURNAMENTS
Revised:  R ANK 2 T OURNAMENTS ⇐⇒ L OCALLY T RANSITIVE T OURNAMENTS

Original: The goal of this section is to characterize the forbidden conﬁgurations of rank 2 tournaments.
Revised: The goal of this section is to characterize the forbidden conﬁgurations of rank 2 tournaments.

Original: Thanks to Lemma 2, this reduces to the problem of identifying a tournament whose representation necessarily span the entire space.
Revised: Thanks to Lemma 2, this reduces to the problem of identifying a tournament whose representation necessarily span the entire space.

Original: The following lemma exhibits this tournament.
Revised: The following lemma exhibits this tournament.

Original: Lemma 3.
Revised: Lemma 3.

Original: Let H =
Revised: Let H =

Original: { h 1 , h 2 , h 3 } ∈ R 2 be two dimensional representation of 3 nodes which induce a 3 cycle tournament.
Revised: { h 1 , h 2 , h 3 } ∈ R 2 be two dimensional representation of 3 nodes which induce a 3 cycle tournament.

Original: Then the set H positively spans R 2 .
Revised: Then the set H positively spans R 2 .

Original: Furthermore, the 3 cycle is the only such tournament on 3 nodes.
Revised: Furthermore, the 3 cycle is the only such tournament on 3 nodes.

Original: The above lemma immediately implies that a coned 3 -cycle is the only forbidden conﬁguration for rank 2 matrices.
Revised: The above lemma immediately implies that a coned 3 -cycle is the only forbidden conﬁguration for rank 2 matrices.

Original: This is indeed true.
Revised: This is indeed true.

Original: However, for the purposes of generalizing our result (which as we will see will be useful when discussing the higher dimension case), we will view the 3 -cycle as a special case of a doubly regular tournament (deﬁned next).
Revised: However, for the purposes of generalizing our result (which as we will see will be useful when discussing the higher dimension case), we will view the 3 -cycle as a special case of a doubly regular tournament (deﬁned next).

Original: Deﬁnition 4.
Revised: Deﬁnition 4.

Original: A tournament T  is said to be  doubly regular if | T + i | =
Revised: A tournament T on n nodes is said to be n - doubly regular if | T + i | =

Original: | T + j | for all i, j and | T + i ∩ 
Revised: | T + j | for all i, j and | T + i ∩ T + j |

Original: T + j | = k for some ﬁxed k for all i (cid:54) =
Revised: = k for some ﬁxed k for all i (cid:54) =

Original: j
Revised: j

Original: Trivially the 3 -cycle is the only 3 -doubly regular tournament.
Revised: Trivially the 3 -cycle is the only 3 -doubly regular tournament.

Original: The following lemma establishes that the ﬂip class of a coned 3 doubly regular tournament only contains itself.
Revised: The following lemma establishes that the ﬂip class of a coned 3 doubly regular tournament only contains itself.

Original: Proposition 2.
Revised: Proposition 2.

Original: The ﬂip class of 3 -doubly-regular-cone does not contain any other tournament.
Revised: The ﬂip class of 3 -doubly-regular-cone does not contain any other tournament.

Original: Proof.

Original: This is easily veriﬁed by checking all tournaments in F ( T ) where T is the coned 3 -cycle.

Original: Theorem 4.
Revised: Theorem 4.

Original: 3 -doubly-regular-cone is the only forbidden conﬁguration for the rank 2 tournament class.
Revised: 3 -doubly-regular-cone is the only forbidden conﬁguration for the rank 2 tournament class.

Original: We have thus far established that any rank 2 tournament on n nodes forbids the 3 -doubly-regular-cone.
Revised: We have thus far established that any rank 2 tournament on n nodes forbids the 3 -doubly-regular-cone.

Original: The advantage of this result is that we can go one step further and explicitly characterize the ranktournament class.
Revised: The advantage of this result is that we can go one step further and explicitly characterize the ranktournament class.

Original: To do this, we need the deﬁnition of a previously studied tournament class.
Revised: To do this, we need the deﬁnition of a previously studied tournament class.

Original: Deﬁnition 5.
Revised: Deﬁnition 5.

Original: A tournament T is called locally transitive if for every node i in T , both T ( T + i ) and
Revised: A tournament T is called locally transitive if for every node i in T , both T ( T + i ) and

Original: T ( T − i ) are transitive tournaments
Revised: T ( T − i ) are transitive tournaments

Original: Before seeing why locally transitive tournaments are relevant to our study, we ﬁrst show that they are intimately connected to transitive tournaments via the following characterization.
Revised: Before seeing why locally transitive tournaments are relevant to our study, we ﬁrst show that they are intimately connected to transitive tournaments via the following characterization.

Original: Theorem 5.
Revised: Theorem 5.

Original: ( Connection between Transitive and Locally Transitive Tournaments )
Revised: ( Connection between Transitive and Locally Transitive Tournaments )

Original: The set of all non isomorphic locally transitive tournaments on n nodes is equivalent to the ﬂip class of the transitive tournament on n nodes.
Revised: The set of all non isomorphic locally transitive tournaments on n nodes is equivalent to the ﬂip class of the transitive tournament on n nodes.

Original: We will see next that this key result allows us to immediately characterize the rank 2 tournament class.
Revised: We will see next that this key result allows us to immediately characterize the rank 2 tournament class.

Original: We will see later (Section 7) that this result is also crucial in determining an upper bound on the dimension needed to represent a given tournament.
Revised: We will see later (Section 7) that this result is also crucial in determining an upper bound on the dimension needed to represent a given tournament.

Original: Theorem 6.
Revised: Theorem 6.

Original: ( Characterization of rank 2 tournaments )
Revised: ( Characterization of rank 2 tournaments )

Original: A tournament T on n nodes is locally transitive if and only if there exists a skew symmetric matric M ∈ R n × n with rank( M )
Revised: A tournament T on n nodes is locally transitive if and only if there exists a skew symmetric matric M ∈ R n × n with rank( M )

Original: = 

Revised: =

Original: such that the T { M } = T .
Revised: such that the T { M } = T .

Original: It is perhaps surprising that a purely structural description of a tournament class namely that of local transitivity turns out to be exactly equivalent to the rank 2 tournament class.
Revised: It is perhaps surprising that a purely structural description of a tournament class namely that of local transitivity turns out to be exactly equivalent to the rank 2 tournament class.

Original: To the best of our knowledge, this characterization appears novel and hasn’t been previously noticed.
Revised: To the best of our knowledge, this characterization appears novel and hasn’t been previously noticed.

Original: One of the interesting consequence of the above characterization is that the minimum feedback arc set problem on rank 2 tournaments can be solved using a standard quick sort procedure.
Revised: One of the interesting consequence of the above characterization is that the minimum feedback arc set problem on rank 2 tournaments can be solved using a standard quick sort procedure.

Original: This is formalized below.
Revised: This is formalized below.

Original: Theorem 7.
Revised: Theorem 7.

Original: ( Minimum Feedback Arc Set is Poly-time Solvable for Rank 2 tournaments ) Let T be a locally transitive tournament on n nodes and let σ 1 be the permutation returned by running a standard quick sort algorithm choosing 1 as the initial pivot node and where the outcome of comparison between items i and j is given according to the sign of T  .
Revised: ( Minimum Feedback Arc Set is Poly-time Solvable for Rank 2 tournaments ) Let T be a locally transitive tournament on n nodes and let σ 1 be the permutation returned by running a standard quick sort algorithm choosing 1 as the initial pivot node and where the outcome of comparison between items i and j is i if and only if i (cid:31) T j .

Original: Let σ k be obtained from σby k − 1 clockwise cyclic shifts for k ∈ [ n ] .
Revised: Let σ k be obtained from σ 1 by k −clockwise cyclic shifts for k ∈ [ n ] .

Original: Let E k be the feedback arc set of σ k w.r.t T .
Revised: Let E k be the feedback arc set of σ k w.r.t T .

Original: Then min k | E k | achieves the minimum size of the feedback arc set for T .
Revised: Then min k | E k | achieves the minimum size of the feedback arc set for T .

Original: Proof.
Revised: Proof.

Original: (Sketch)
Revised: (Sketch)

Original: The proof involves two steps: ﬁrst arguing that by ﬁxing any pivot, quick sort would return a ranking that is a cyclic shift of σ 1 .
Revised: The proof involves two steps: ﬁrst arguing that by ﬁxing any pivot, quick sort would return a ranking that is a cyclic shift of σ 1 .

Original: The second step involves inductively arguing that one of the cyclic shift must necessarily minimize the feedback arc set.
Revised: The second step involves inductively arguing that one of the cyclic shift must necessarily minimize the feedback arc set.

Original: 5. R ANK 4 T OURNAMENTS
Revised: 5. R ANK 4 T OURNAMENTS

Original: We now turn to rank 4 tournaments.
Revised: We now turn to rank 4 tournaments.

Original: We could have directly considered rank d tournaments, but it turns out that what we can show a slightly stronger result for rank 4 tournaments than the general case and so we focus on them separately.
Revised: We could have directly considered rank d tournaments, but it turns out that what we can show a slightly stronger result for rank 4 tournaments than the general case and so we focus on them separately.

Original: While it is arguably simple in the rank 2 case to identify the tournament that necessitates the positive spanning property, it is not immediately clear in the rankcase.
Revised: While it is arguably simple in the rank 2 case to identify the tournament that necessitates the positive spanning property, it is not immediately clear in the rankcase.

Original: A ﬁrst guess would be to consider the regular tournaments (as the 3 cycle for rank 2 is also a regular tournament) on 5 nodes or 7 nodes.
Revised: A ﬁrst guess would be to consider the regular tournaments (as the 3 cycle for rank 2 is also a regular tournament) on 5 nodes or 7 nodes.

Original: However, these turn out to be insufﬁcient as one can construct counter examples of regular tournaments on up to 7 nodes with representations that don’t span the entire R 4 .
Revised: However, these turn out to be insufﬁcient as one can construct counter examples of regular tournaments on up to 7 nodes with representations that don’t span the entire R 4 .

Original: In fact, as we had deﬁned earlier, the right way to generalize to higher dimension turns out to be using doubly regular tournaments.
Revised: In fact, as we had deﬁned earlier, the right way to generalize to higher dimension turns out to be using doubly regular tournaments.

Original: Theorem 8.
Revised: Theorem 8.

Original: The 11 -doubly-regular-cone is a forbidden conﬁguration for rank 4 tournament class.
Revised: The 11 -doubly-regular-cone is a forbidden conﬁguration for rank 4 tournament class.

Original: Note that while for the rank 2 case, we were able to prove that the only forbidden ﬂip class is the one that contains a coned 3 cycle, we have not shown that the only forbidden conﬁguration for rankclass is the 12 doubly regular cone.
Revised: Note that while for the rank 2 case, we were able to prove that the only forbidden ﬂip class is the one that contains a coned 3 cycle, we have not shown that the only forbidden conﬁguration for rankclass is the 11 doubly regular cone.

Original: In fact, we believe that the smallest forbidden tournament for rank 4 class is the 7 - doubly regular tournament.
Revised: In fact, we believe that the smallest forbidden tournament for rank 4 class is the 7 - doubly regular cone.

Original: However, we haven’t been able to prove this.
Revised: However, we haven’t been able to prove this.

Revised: This appears to be a non-trivial problem.

Revised: From our simulation experiments, we observe that the only ﬂip class that could be forbidden on 8 nodes is the one that contains the 7 -doubly regular cone.

Revised: In particular, we were able to produce examples of representations for all other ﬂip classes on 8 nodes.

Revised: However, this does not imply that the ones where we could not produce a forbidden conﬁguration is in fact forbidden.

Revised: Unfortunately, it seems tricky to prove this and we don’t have a way to show this at this point.

Original: On the other hand, as we will see in the next section, the result in Theorem 8 is still stronger than the result for the general rank d tournaments.
Revised: On the other hand, as we will see in the next section, the result in Theorem 8 is still stronger than the result for the general rank d tournaments.

Original: Having discussed rank 2 and rank 4 cases separately, we next turn our attention to the general rank d tournament class.
Revised: Having discussed rank 2 and rank 4 cases separately, we next turn our attention to the general rank d tournament class.

Original:  R ANK d TOURNAMENTS AND THE H ADAMARD C ONJECTURE
Revised:  R ANK d TOURNAMENTS AND THE H ADAMARD C ONJECTURE

Original: From the understanding of rank 2 and rank 4 tournament classes in the previous sections, and noting that the corresponding forbidden conﬁgurations are intimately related to doubly regular tournaments, it is tempting to conjecture that this is true in general.
Revised: From the understanding of rank 2 and rank 4 tournament classes in the previous sections, and noting that the corresponding forbidden conﬁgurations are intimately related to doubly regular tournaments, it is tempting to conjecture that this is true in general.

Original: Conjecture 1.
Revised: Conjecture 1.

Original: Rank 2 d tournament class forbids 4 d − 1 -doubly-regular-cones.
Revised: Rank 2 d tournament class forbids 4 d − 1 -doubly-regular-cones.

Original: Ideally, the conjecture above must have a qualiﬁer ‘if they exist’ for the 4 d − 1 doubly regular cone.
Revised: Ideally, the conjecture above must have a qualiﬁer ‘if they exist’ for the 4 d − 1 doubly regular cone.

Original: This is because of the equivalence between doubly regular tournaments on 4 d − 1 nodes and
Revised: This is because of the equivalence between doubly regular tournaments on 4 d − 1 nodes and

Original: Hadamard matrices in { +1 , − 1 } 4 d × 4 d (Reid & Brown (1972)).
Revised: Hadamard matrices in { +1 , − 1 } 4 d × 4 d (Reid & Brown (1972)).

Original: A matrix H ∈ { +1 , − 1 } n × n is called Hadamard if H (cid:48)
Revised: A matrix H ∈ { +1 , − 1 } n × n is called Hadamard if H (cid:48)

Original: H =
Revised: H =

Original: nI where I is the identity matrix.
Revised: nI where I is the identity matrix.

Original: It is known that there is a bijection between skew Hadamard matrices and doubly regular tournaments Reid & Brown (1972).
Revised: It is known that there is a bijection between skew Hadamard matrices and doubly regular tournaments Reid & Brown (1972).

Original: A long standing unsolved conjecture about Hadamard matrices is the following:
Revised: A long standing unsolved conjecture about Hadamard matrices is the following:

Original: Conjecture 2.
Revised: Conjecture 2.

Original: (Hadamard) There exists a Hadamard matrix of order 4 d for every d > 0 .
Revised: (Hadamard) There exists a Hadamard matrix of order 4 d for every d > 0 .

Original: If Conjecture 1 were true, then it would imply the existence of 4 d − 1 -doubly regular tournament for every d and thus would imply the Hadamard conjecture is true.
Revised: If Conjecture 1 were true, then it would imply the existence of 4 d − 1 -doubly regular tournament for every d and thus would imply the Hadamard conjecture is true.

Original: In fact, Conjecture 1 being true would say more which we state below:
Revised: In fact, Conjecture 1 being true would say more which we state below:

Original: Conjecture 3.
Revised: Conjecture 3.

Original: There exists a skew symmetric Hadamard matrix of order 4 d for every d > 0 .
Revised: There exists a skew symmetric Hadamard matrix of order 4 d for every d > 0 .

Original: The main result of this section is a weaker form of the conjecture:
Revised: The main result of this section is a weaker form of the conjecture:

Original: Theorem 9.
Revised: Theorem 9.

Original: Rank 2( d − 1) tournament class forbids 4 d 2 − 1 -doubly-regular-cones if they exist.Proof.
Revised: Rank 2( d − 1) tournament class forbids 12 d 2 − 1 -doubly-regular-cones if they exist.

Original: The result follows the arguments in Forster & Simon (2006).

Original: We note that the arguments in Forster & Simon (2006) work only for non-zero sign matrices.

Original: By overloading T to denote the signed adjacency matrix of the corresponding tournament, we can consider the non-zero sign matrix

Original: G = T

Original: + I where I ∈ R n × n is the identity matrix.

Original: It is easy to see that

Original: [-ρ i ( G ) 
-]
Revised: [-ρ i ( G ) 
-]

Original: = ρ i ( T ) 2 + 1 where ρ i denotes the i -th largest singular value of the corresponding matrix.

Original: It then follows from Forster &

Original: Simon (2006) that if G has an representation in dim dimensions, then it must be the case that dim

Original: [Equation]

Original: Noting that for a doubly regular cone T on n nodes, ρ i ( T ) .

Original: 2 = n − 1 ∀

Original: i , we get dim

Original: [Equation]

Original: n

Original: Thus to get a matrix M such that sign ( M ) =

Original: G , one needs at least √ n dimensions i.e., rank ( M ) ≥ √ n .

Original: However, this does not immediately imply that M is skew symmetric.

Original: But one can consider the matrix M − M T which has the same sign pattern as M except for the diagonals which are 0 .

Original: But this is exactly the skew symmetric matrix that we need to represent T .

Original: Note that the rank( M − M T ) is at most 2 dim .

Original: As we can convert any representation in dim for G to a skew symmetric embedding for

Original: T in at most 2 dim dimensions, the lower bound of √ n applies for the tournaments as well.

Original: Setting n = 4 d 2 to the number of nodes in the doubly regular cone as given in the Theorem, we get that at least 2 d dimensions are needed to embed such a tournament.

Original: The result follows.

Original:  H OW M ANY D IMENSIONS ARE NEEDED TO R EPRESENT A T OURNAMENT ?
Revised:  H OW M ANY D IMENSIONS ARE NEEDED TO R EPRESENT A T OURNAMENT ?

Original: The previous sections considered a speciﬁc rank d tournament class and tried to characterize them using forbidden conﬁgurations.
Revised: The previous sections considered a speciﬁc rank d tournament class and tried to characterize them using forbidden conﬁgurations.

Original: In this section, We turn to the dual question of understanding the minimum number of dimensions needed to embed a tournament.
Revised: In this section, We turn to the dual question of understanding the minimum number of dimensions needed to embed a tournament.

Original: We start by not considering a single tournament T but the set of all tournaments on n nodes.
Revised: We start by not considering a single tournament T but the set of all tournaments on n nodes.

Original: We show below a general result which provides a lower bound on the minimum dimension needed to embed any tournament on n nodes.
Revised: We show below a general result which provides a lower bound on the minimum dimension needed to embed any tournament on n nodes.

Original: Theorem 10.
Revised: Theorem 10.

Original: (Lower Bound on minimum representation dimension) Let T be a tournament on n nodes.
Revised: (Lower Bound on minimum representation dimension) Let T be a tournament on n nodes.

Original: Then there exists H
Revised: Then there exists H

Original: = { h 1 , . . .
Revised: = { h 1 , . . .

Original: , h n } ∈ R d such that T =
Revised: , h n } ∈ R d such that T =

Original: T [ H ] only if d (cid:80) ni
Revised: T [ H ] only if d (cid:80) ni

Original: =1 ( ρ i ( T + I ))
Revised: =1 ( ρ i ( T + I ))

Original: 2 ≥ n 2 .
Revised: 2 ≥ n 2 .

Original: Furthermore, let d be the minimum dimension needed to embed every tournament on n nodes.
Revised: Furthermore, let d be the minimum dimension needed to embed every tournament on n nodes.

Original: Then, d ≥ √ n .
Revised: Then, d ≥ √ n .

Original: Proof.

Original: The proof is exactly the same as the ﬁrst part of the proof of Theorem 9.

Original: The result follows the arguments in the celebrated work of Forster & Simon (2006).
Revised: The result follows the arguments in the celebrated work of Forster & Simon (2006).

Original: As Alon et al.
Revised: As Alon et al.

Original: (2016) point out, Forster’s technique cannot be stretched further in obvious ways to get upper bounds.
Revised: (2016) point out, Forster’s technique cannot be stretched further in obvious ways to get upper bounds.

Original: The above theorem tells us that in the worst case at least O ( √ n ) dimensions are necessary to represent all tournaments on n nodes.
Revised: The above theorem tells us that in the worst case at least Ω( √ n ) dimensions are necessary to represent all tournaments on n nodes.

Original: In fact if Conjecture 1 were true, the minimum dimension would be O ( n ) .
Revised: In fact if Conjecture 1 were true, the minimum dimension would be Ω( n ) .

Original: However, in practice one might not encounter tournaments with such extremal/worst-case properties.
Revised: However, in practice one might not encounter tournaments with such extremal/worst-case properties.

Revised: Remark: The lower bound for the representation dimension of random tournaments (where the orientation of each edge is determined by an unbiased Bernoulli coin toss) will depend on the singular values of random tournaments.

Revised: However, we don’t expect the representation dimension to be of independent of n , the number of nodes.

Revised: Loosely speaking, a doubly regular tournament is "like" a random tournament (the associated Hadamard Tournament has been used in deterministic perturbation schemes as alternatives for random sign matrices (Bhatnagar et al. (2003))).

Revised: On the other hand, the most interesting real-world tournaments might be characterized by constant sized node representations and hence may be structurally much more constrained than random tournaments.

Original: Typically, a smaller number of dimensions might be enough to represent tournaments of practical interest.
Revised: Typically, a smaller number of dimensions might be enough to represent tournaments of practical interest.

Original: Our goal below is to upper bound on the number of dimensions needed to embed a given tournament T .
Revised: Our goal below is to upper bound on the number of dimensions needed to embed a given tournament T .

Original: Recall that E σ ( T ) denotes the feedback arc set of a permutation σ w.r.t a tournament T .
Revised: Recall that E σ ( T ) denotes the feedback arc set of a permutation σ w.r.t a tournament T .

Original: We deﬁne the number of nodes involved in the feedback arc set as follows:
Revised: We deﬁne the number of nodes involved in the feedback arc set as follows:

Original: θ ( σ, T ) =
Revised: θ ( σ, T ) =

Original: |{ i : ∃
Revised: |{ i : ∃

Original: j : σ ( i ) > σ ( j ) , ( i, j ) ∈
Revised: j : σ ( i ) > σ ( j ) , ( i, j ) ∈

Original: E σ ( T ) }| .
Revised: E σ ( T ) }| .

Original: We next deﬁne a crucial quantity µ ( T ) which we term as the Flip Feedback Node set size .
Revised: We next deﬁne a crucial quantity µ ( T ) which we term as the Flip Feedback Node set size .

Original: This quantity will determine an upper bound on the dimension where a tournament can be represented:
Revised: This quantity will determine an upper bound on the dimension where a tournament can be represented:

Revised: Deﬁnition 6.

Revised: Given a tournament T , deﬁne the Flip Feedback Node Set Size as follows:

Original: [Equation]
Revised: [Equation]

Original: In words, given a tournament T , the quantity µ ( T ) captures the minimum number of nodes involved in any feedback arc set among all tournaments in the ﬂip class of T .
Revised: In words, given a tournament T , the quantity µ ( T ) captures the minimum number of nodes involved in any feedback arc set among all tournaments in the ﬂip class of T .

Original: For instance if T is a locally transitive tournament then µ ( T ) would be 0 - as T is necessarily in the ﬂip class of a transitive tournament and so the E σ corresponding to the topological ordering of the transitive tournament would have an empty feedback arc set.
Revised: For instance if T is a locally transitive tournament then µ ( T ) would be 0 - as T is necessarily in the ﬂip class of a transitive tournament and so the E σ corresponding to the topological ordering of the transitive tournament would have an empty feedback arc set.

Original: As another example, consider T to be the coned 3 -cycle.
Revised: As another example, consider T to be the coned 3 -cycle.

Original: The ﬂip class of this tournament contains only itself and the best permutation will have one edge in the feedback arc set.
Revised: The ﬂip class of this tournament contains only itself and the best permutation will have one edge in the feedback arc set.

Original: Thus µ ( T ) = 1 .
Revised: Thus µ ( T ) = 1 .

Original: In general, it is trivially true that µ ( T ) is upper bounded by n , the number of nodes.
Revised: In general, it is trivially true that µ ( T ) is upper bounded by n , the number of nodes.

Original: However µ ( T ) could be much smaller than n depending on T .
Revised: However µ ( T ) could be much smaller than n depending on T .

Original: The main result of this section is the theorem below that shows that µ ( T ) gives an upper bound on the number of dimension needed to represent any tournament.
Revised: The main result of this section is the theorem below that shows that µ ( T ) gives an upper bound on the number of dimension needed to represent any tournament.

Original: Theorem 11.
Revised: Theorem 11.

Original: (Upper Bound on minimum representation dimension)
Revised: (Upper Bound on minimum representation dimension)

Original: For any tournament T , there exists H
Revised: For any tournament T , there exists H

Original: = { h 1 , . . .
Revised: = { h 1 , . . .

Original: , h n } ∈ R 2( µ ( T )+1)
Revised: , h n } ∈ R 2( µ ( T )+1)

Original: such that T = T [ H ] .
Revised: such that T = T [ H ] .

Original: Proof.

Original: Given a tournament T , we ﬁrst show that we can start with an arbitrary transitive tournament and add enough rank 2 corrections to obtain a representation for T .

Original: Every addition of a rank 2 matrix will increase the representation dimension by at most 2 .

Original: The result will follow then by noting that for the choice of the transitive tournament which minimizes the number of corrections needed, one needs at most 2( µ ( T ) + 1) representation dimension.

Original: Let T (cid:48) be an arbitrary transitive tournament which has a 2 dimensional representation and let

Original: M ∈ R n × n be the associated skew symmetric matrix that represents T (cid:48) .

Original: W.l.o.g, assume that

Original: M ij > 0 if and only if i < j .

Original: Deﬁne E k ( T )

Original: =

Original: { i : i < k, i < T k } .

Original: By deﬁnition, the feedback arc set E σ ( T ) =

Original: ∪ nk =1 ( k, E k ( T ))

Original: where σ = [1 , . . .

Original: , n ] is the topological order corresponding to the transitive tournament T (cid:48) .

Original: We start with k = n and correct the feedback arc errors arising from E k iteratively.

Original: Deﬁne ∆ n = (cid:113) max i<n ( M in + (cid:15) ) for some small enough (cid:15) > 0 .

Original: Let u ∈ R n be such that u i =

Original: [-∆ ∀ i ∈ E n and u i = 
-]
Revised: [-∆ ∀ i ∈ E n and u i = 
-]

Original: otherwise.

Original: Let v ∈ R n be such that v n = − ∆ , v i = 0 ∀ i (cid:54) = n .

Original: [-It is easy to verify that M + uv T − vu T represents a tournament that has the feedback arc set ∪ n − 1 k =
-]
Revised: [-It is easy to verify that M + uv T − vu T represents a tournament that has the feedback arc set ∪ n − 1 k =
-]

Original: ( k, E k ( T ))

Original: i.e., the errors in E n have been corrected.

Original: The cost of correcting the error is adding a rank 2 skew symmetric matrix which increases the representation dimension by at most 2 .

Original: One can repeat the same procedure for n − 1 , n − 2 , . . .

Original: until all errors are corrected.

Original: The upper bound in the theorem follows noting that the above argument works for any transitive tournament T (cid:48) and so we can start with the one which has the least number of nodes involved in the feedback arc set to minimize the number of extra dimensions needed to represent T .

Original: Remark 1: The above theorem says that one can always obtain an representation H in dimension d
Revised: Remark 1: The above theorem says that one can always obtain an representation H in dimension d

Original: = O ( µ ( T )) that realizes T .
Revised: = O ( µ ( T )) that induces T .

Original: The bound gets tighter for tournaments with smaller feedback arc sets, which is what one might typically expect in practice.
Revised: The bound gets tighter for tournaments with smaller feedback arc sets, which is what one might typically expect in practice.

Original: Note that even for some tournaments that may  have a large feedback arc set, the associated ﬂip class might contain a tournament with a smaller feedback arc set.
Revised: Note that even for some tournaments that may [Figure] have a large feedback arc set, the associated ﬂip class might contain a tournament with a smaller feedback arc set.

Original: Remark 2: We note that in general µ ( T ) is not necessarily the cardinality of the minimum feedback arc set among all tournaments in the ﬂip class of T .
Revised: Remark 2: We note that in general µ ( T ) is not necessarily the cardinality of the minimum feedback arc set among all tournaments in the ﬂip class of T .

Original: Instead µ ( T ) captures the cardinality of the[Figure] set of nodes involved in any feedback arc set.
Revised: Instead µ ( T ) captures the cardinality of the set of nodes involved in any feedback arc set.

Original: To see why these two could be different, consider the tournaments in Figure 2.
Revised: To see why these two could be different, consider the tournament in Figure 2.

Original: Here, σ a = [6 1 2 3 4 5] is the permutation minimizing the feedback arc set.
Revised: Here, σ a = [6 1 2 3 4 5] is the permutation minimizing the feedback arc set.

Original: Let σ b =
Revised: Let σ b =

Original: Note that | E σ a | = 2 , θ ( σ a , T ) =
Revised: Note that | E σ a | = 2 , θ ( σ a , T ) =

Original: |{ 1 , 2 }| = 2 .
Revised: |{ 1 , 2 }| = 2 .

Original: However | E σ b | = 4 , yet θ ( σ b , T ) =
Revised: However | E σ b | = 3 , yet θ ( σ b , T ) =

Original: |{ 6 }| = 1 .
Revised: |{ 6 }| = 1 .

Original: Thus, σ b gives a tighter upper bound on the dimension needed to represent T .
Revised: Thus, σ b gives a tighter upper bound on the dimension needed to represent T .

Original:  7. C ONNECTIONS TO S IGN R ANK
Revised:  A PPLICATIONS O F T HE R ESULTS 8. C ONNECTIONS TO S IGN R ANK

Original: The sign rank of a matrix G ∈ { +1 , − 1 } m × n is deﬁned as the smallest integer d such that there exists a matrix M ∈ R m × n of rank d that satisﬁes sign ( M )
Revised: The sign rank of a matrix G ∈ { +1 , − 1 } m × n is deﬁned as the smallest integer d such that there exists a matrix M ∈ R m × n of rank d that satisﬁes sign ( M )

Original: =
Revised: =

Original: G 1 .
Revised: G 1 .

Original: Here sign ( z ) = 

Revised: Here sign ( z ) =

Original: if z >and − 1 otherwise.
Revised: if z >and − 1 otherwise.

Original: A breakthrough result on the lower bound on the sign rank was given by Forster & Simon (2006).
Revised: A breakthrough result on the lower bound on the sign rank was given by Forster & Simon (2006).

Original: However, good upper bounds have been harder to obtain.
Revised: However, good upper bounds have been harder to obtain.

Original: We show below how
Revised: We show below how

Original: Theorem 11 also translates to an upper bound on the sign rank of any sign matrix G .
Revised: Theorem 11 also translates to an upper bound on the sign rank of any sign matrix G .

Original: Theorem 12.
Revised: Theorem 12.

Original: Let G ∈ { +1 , − 1 } m × n be an arbitrary sign matrix.
Revised: Let G ∈ { +1 , − 1 } m × n be an arbitrary sign matrix.

Original: Let T be any tournament on m + n nodes.
Revised: Let T be any tournament on m + n nodes.

Original: Let S = { 1 , . . .
Revised: Let S = { 1 , . . .

Original: , m } and let edge orientations of the cut( S, ¯ S ) in T be determined by
Revised: , m } and let edge orientations of the cut( S, ¯ S ) in T be determined by

Original: G Then sign-rank( G ) ≤ 2( µ ( T ) + 1)
Revised: G Then sign-rank( G ) ≤ 2( µ ( T ) + 1)

Original: Proof.
Revised: We argue that the the above result can give signiﬁcantly tigher bounds than the bounds in Alon et al.

Original: From Theorem 11, T can be represented using at most 2( µ ( T ) + 1) dimensions.
Revised: The simplest example one can consider is a locally transitive tournament on n nodes.

Original: By construction any representation of T must also represent G .
Revised: Viewing this as a sign pattern matrix, it is easy to argue that the VC dimension of such a matrix is atmost 2 .

Original: The result follows.
Revised: Theorem 5 in Alon et al. (2016) shows that the sign rank is upper bounded by O ( √ n )) when VC dimension is 2 (For a general hypothesis class of VC dimension d , the upper bound is O ( n 1 − 1 d ) . On the other hand, by deﬁnition µ ( T ) for any locally transitive tournament is exactly 0 (and so the upper bound of 2 µ ( T ) + 1 is exactly 2 and is tight) for any locally transitive tournament which is independent of the number of nodes n .

Original: As far as we know, the previous known upper bounds for  sign rank depended either on the VCdimension of the natural binary function class associated with G or assumed extra regularity conditions (for instance see Alon et al. (2016) for upper bounds of ∆ regular sign pattern matrices).
Revised: The important reason for this signiﬁcantly improved bound using µ ( T ) is that we consider only skew symmetric sign pattern matrices while Alon et al. (2016) look at the worst case (w.r.t representation dimension/sign rank) sign pattern matrices for a ﬁxed VC dimension.

Original: Our bound reduces the study of sign rank to a more graph theoretic study of the feedback arc set problem.
Revised: Our bound reduces the study of sign rank to a more graph theoretic study of the feedback arc set problem.

Original: It is not clear if the upper bound of 2( µ ( T ) + 1) can be improved and we leave this to future work.
Revised: It is not clear if the upper bound of 2( µ ( T ) + 1) can be improved and we leave this to future work.

Original: 7. R EAL W ORLD T OURNAMENT E XPERIMENTS
Revised: 8. C ONNECTIONS T O L EARNING FROM P AIRWISE C OMPARISONS

Revised: Consider a learning to rank problem from pairwise comparisons.

Revised: Here, a set of n items need to be ranked from a subset of pairwise comparisons among them.

Revised: Every pair that is chosen for comparison is compared a ﬁxed number of times.

Revised: Every time items i and j are compared, i is preferred over j with probability P

Revised: i

Revised: j .

Revised: A common and popular model to capture these probabilities is the BradleyTerry-Luce (BTL) model Bradley & Terry (1952) where P i j

Revised: = s i /

Revised: ( s i + s j ) for some score vector s ∈ R n .

Revised: Note that the model is completely speciﬁed by the score vector s which in turn completely determines the probability preference matrix P .

Revised: The learning problem is to learn these unknown score vector s ∈ R n from pairwise comparisons.

Revised: Once the score vector is learnt, a ranking can be obtained by sorting these scores and the pairwise probabilities of unseen pairs of items can be predicted.

Revised: The above BTL model is an example of a rank- 2 model in the sense that the probability preference matrix P results in a rank 2 matrix under the log-odds skew symmetric transformation.

Revised: Indeed, if we deﬁne M ij = log( P ij P ji ) .

Revised: then equivalently M i

Revised: j = log( s i ) − log( s j ) .

Revised: It is easy to show that M is a rank 2 skew symmetric matrix.

Revised: However, the major disadvantage of the BTL model is that it can capture only transitive preferences i.e, P ij > 0 .

Revised: 5 and P jk > 0 .

Revised: 5 =

Revised: ⇒ P ki > 0 .

Revised: In real world situations, intransitivity is very common.

Revised: To achieve intransitivity, the simplest way would be to start with a general rank r skew symmetric matrix M and consider the probability matrix that determines the preference probabilities as P ij = 1 / (1 + exp( − M ij ) .

Revised: Here, the learning problem would be to estimate the two score vectors or equivalently the two dimensional representation for each item.

Revised: Previous studies (Rajkumar & Agarwal (2016)) show that this can be learnt using Matrix completion based approaches or maximum likelihood based approaches (Chen & Joachims (2016)).

Revised: However, what was not known earlier is the structure of tournaments that can be captured using such low rank restrictions which is important to decide on the parameter r .

Revised: Our work helps modellers make informed decision to choose r based on explicitly identifying the types of tournaments that can be modelled and subsequently learnt.

Revised: We discuss next a simple application where modeling a ranking from pairwise comparisons problem can beneﬁt from the insights gained using our results.

Revised: Consider a situation of modelling sports tournaments such as Tennis.

Revised: Here, one can choose to model the players (nodes) using 2 -dimensions where the dimensions corresponds to their offense (forehand) and defense (backhand) strengths respectively.

Revised: When two players compete, the advantage of the offense strength of player 1 w.r.t the defense of player 2 and vice versa determine the outcome of the match.

Revised: This is precisely captured by a rank 2 model where the learning problem would be to infer these latent offense/defense strength of each player from outcomes of pairwise competitions.

Revised: Theorem 6 says that such a model would immediately lead to locally transitive tournaments among the players.

Revised: This structural characterisation now gives insights to the modeller if 2 dimensions are enough to model the players or not.

Revised:  R EAL W ORLD T OURNAMENT E XPERIMENTS

Original: We conducted simple experiments on real world data sets.
Revised: We conducted simple experiments on real world data sets.

Original: Speciﬁcally, we considered 114 real world tournaments that arise in several applications including election candidate preferences, Sushi preferences, cars preferences, etc (source: www.preflib.org ).
Revised: Speciﬁcally, we considered 114 real world tournaments that arise in several applications including election candidate preferences, Sushi preferences, cars preferences, etc (source: www.preflib.org ).

Original: The number of nodes in these tournaments varied from 5 to 23 .
Revised: The number of nodes in these tournaments varied from 5 to 23 .

Original: Out of the 114 tournaments considered, 76(66 .
Revised: Out of the 114 tournaments considered, 76(66 .

Original: 67%) were in fact locally transitive.
Revised: 67%) were in fact locally transitive.

Original: For these tournaments, the upper bounds and lower bounds given by our theorems matched and was equal to 2 .
Revised: For these tournaments, the upper bounds and lower bounds given by our theorems matched and was equal to 2 .

Original: Interestingly, even for the non-locally transitive tournaments, the lower bound still turned out to be 2 .
Revised: Interestingly, even for the non-locally transitive tournaments, the lower bound still turned out to be 2 .

Original: We computed the upper bound for tournaments of size at most 9 (we did not do it for larger tournaments as this involves a brute force search) and found the value to be either 4 or 6 .
Revised: We computed the upper bound for tournaments of size at most 9 (we did not do it for larger tournaments as this involves a brute force search) and found the value to be either 4 or 6 .

Original: This shows that the upper bounds are usually non-trivial and efﬁciently approximating it is an interesting direction for future work.
Revised: This shows that the upper bounds are usually non-trivial and efﬁciently approximating it is an interesting direction for future work.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: In this work, we develop a theory of tournament representations.
Revised: In this work, we develop a theory of tournament representations.

Original: We show how ﬁxing the representation dimension enforces, via forbidden conﬁgurations, restrictions on the type of tournaments that can be represented.
Revised: We show how ﬁxing the representation dimension enforces, via forbidden conﬁgurations, restrictions on the type of tournaments that can be represented.

Original: We study and characterize rank 2 tournaments and show forbidden sub-tournaments for the rank d tournament class.
Revised: We study and characterize rank 2 tournaments and show forbidden sub-tournaments for the rank d tournament class.

Original: We develop upper and lower bounds for minimum dimension needed to represent a tournament.
Revised: We develop upper and lower bounds for minimum dimension needed to represent a tournament.

Original: Future work includes attempting to look deeper into some of the conjectures presented and possible strengthening of some of the bounds presented.
Revised: Future work includes attempting to look deeper into some of the conjectures presented and possible strengthening of some of the bounds presented.

Original: A A PPENDIX Proof of Lemma
Revised: A A PPENDIX Proof of Theorem

Original: Proof.
Revised: Proof.

Revised: Assume there is a T (cid:48) ∈ F ( T ) which is not forbidden for rank d tournament class.

Revised: Thus, ∃ H (cid:48)

Revised: = { h (cid:48) 1 , . . .

Revised: , h (cid:48) n } ∈ R d such that T (cid:48)

Revised: =

Revised: T [ H (cid:48) ] .

Revised: By deﬁnition, there must also exist a S ⊆

Revised: [ n ] such that T (cid:48)

Revised: =

Revised: φ S ( T ) .

Revised: Consider the representation H obtained from H (cid:48) where h i = − h (cid:48)

Revised: i for all i ∈ S and h i = h (cid:48)

Revised: i otherwise.

Revised: It is easy to verify that T

Revised: = T [ H ] which is a contradiction to the assumption that T is a forbidden conﬁguration for rank d tournaments.

Revised: Proof of Proposition

Original: Wlog, assume that 1 ≥ T 2 ≥ T 3 ≥ T 1 .
Revised: Proof. Consider any tournament T . Let T (cid:48) = φ { i ∪ T − i } ( T ) for an arbitrary node i .

Revised: By deﬁnition

Revised: T (cid:48) ∈ F ( T ) .

Revised: Also, T (cid:48) is a R -cone, coned by i .

Revised: Proof of Lemma

Revised: Proof.

Revised: Consider any representation H that induces a tournament R .

Revised: By assumption of the theorem,

Revised: H positively spans R d .

Revised: Thus, by Farkas’ lemma, there cannot exist a v ∈ R d such that v T h i > 0 ∀ i .

Revised: Note that if R -cone is not a forbidden conﬁguration for rank d tournaments, then there must exist a h ∈ R d such that h T A rot h i > 0 ∀ i .

Revised: As A rot is invertible, one can set v =

Revised: ( A rot ) T h with the property that v T h

Revised: i > 0 ∀ i .

Revised: But this contradicts the conclusion drawn earlier from Farka’s lemma.

Revised: Proof of Proposition

Revised: Proof.

Revised: This is easily veriﬁed by checking all tournaments in F ( T ) where T is the coned 3 -cycle.

Revised: Proof of Lemma

Original:  Then, it must be the case that the counterclockwise angle between the representation of the corresponding items must be ≤ 180 degrees.
Revised: Proof. Wlog, assume that 1 ≥ T 2 ≥ T 3 ≥ T 1 . Then, it must be the case that the counterclockwise angle between the representation of the corresponding items must be ≤ 180 degrees.

Original: However, if the representations { h 1 , h 2 , h 3 } did not positively span R 2 , then by Farka’s Lemma, there must be some supporting hyperplane for the representations.
Revised: However, if the representations { h 1 , h 2 , h 3 } did not positively span R 2 , then by Farka’s Lemma, there must be some supporting hyperplane for the representations.

Original: However, this would imply at least one of the node pairs { (1 , 2) , (2 , 3) , (3 , 1) } must necessarily make an angle ≥ 180 degrees.
Revised: However, this would imply at least one of the node pairs { (1 , 2) , (2 , 3) , (3 , 1) } must necessarily make an angle ≥ 180 degrees.

Original: But this contradicts the assumption that the nodes form a 3 cycle.
Revised: But this contradicts the assumption that the nodes form a 3 cycle.

Original: Proof of Theorem
Revised: Proof of Theorem

Original: Proof.
Revised: Proof.

Original: Let T be a transitive tournament on n nodes and let T (cid:48) ∈ F ( T ) .
Revised: Let T be a transitive tournament on n nodes and let T (cid:48) ∈ F ( T ) .

Original: Then, there exists some
Revised: Then, there exists some

Original: S ⊆
Revised: S ⊆

Original: [ n ] such that T (cid:48)
Revised: [ n ] such that T (cid:48)

Original: =
Revised: =

Original: φ S ( T ) .
Revised: φ S ( T ) .

Original: Consider any node i ∈ [ n ] .
Revised: Consider any node i ∈ [ n ] .

Original: Deﬁne following 4 subset of nodes associated with i : S 1 = T + i ∩ S, S 2 =
Revised: Deﬁne following 4 subset of nodes associated with i : S 1 = T + i ∩ S, S 2 =

Original: T + i \ S 1 , S 3 =
Revised: T + i \ S 1 , S 3 =

Original: T − i ∩ S, S 4 =
Revised: T − i ∩ S, S 4 =

Original: T − i \ S .
Revised: T − i \ S .

Original: Note that each of
Revised: Note that each of

Original: T ( S k ) for k = 1 to 4 is a transitive sub-tournament and the relationship across S i , S j for any two sets is either one completely beats the other or completely loses to the other.
Revised: T ( S k ) for k = 1 to 4 is a transitive sub-tournament and the relationship across S i , S j for any two sets is either one completely beats the other or completely loses to the other.

Original: Note that in φ S ( T ) the orientation of the edges across these sets is either ﬂipped as a whole or not ﬂipped at all.
Revised: Note that in φ S ( T ) the orientation of the edges across these sets is either ﬂipped as a whole or not ﬂipped at all.

Original: Thus, exactly two of these sets will be part of T (cid:48)
Revised: Thus, exactly two of these sets will be part of T (cid:48)

Original: + i and two part of T − i (the exact sets among these sets will depend on whether i ∈ S or not), thus preserving the local transitivity property.
Revised: + i and two part of T − i (the exact sets among these sets will depend on whether i ∈ S or not), thus preserving the local transitivity property.

Original: Thus T (cid:48) is locally transitive.
Revised: Thus T (cid:48) is locally transitive.

Original: To prove the opposite direction, let T be a locally transitive tournament.
Revised: To prove the opposite direction, let T be a locally transitive tournament.

Original: Let i ∈ [ n ] be an arbitrary node.
Revised: Let i ∈ [ n ] be an arbitrary node.

Original: We argue that T (cid:48) :
Revised: We argue that T (cid:48) :

Original: = φ T + i ( T ) is a transitive tournament.
Revised: = φ T + i ( T ) is a transitive tournament.

Original: We will show this by arguing that there does not exist a 3 cycle in T (cid:48) .
Revised: We will show this by arguing that there does not exist a 3 cycle in T (cid:48) .

Original: Consider any 3 -cycle a > T b > T c > T a .
Revised: Consider any 3 -cycle a > T b > T c > T a .

Original: As T is locally transitive, not all { a, b, c } can be in T
Revised: As T is locally transitive, not all { a, b, c } can be in T

Original: + i .
Revised: + i .

Original: Also not all { a, b, c } can avoid T
Revised: Also not all { a, b, c } can avoid T

Original: + i as [ n ] \ T
Revised: + i as [ n ] \ T

Original: + i is transitive.
Revised: + i is transitive.

Original: Thus at least one and at most two of { a, b, c } belongs to T
Revised: Thus at least one and at most two of { a, b, c } belongs to T

Original: + i ( T ) .
Revised: + i ( T ) .

Original: This means that the 3 -cycle becomes a transitive tournament in T (cid:48) :
Revised: This means that the 3 -cycle becomes a transitive tournament in T (cid:48) :

Original: = φ T + i ( T ) .
Revised: = φ T + i ( T ) .

Original: Thus every cycle in T becomes transitive in T (cid:48) .
Revised: Thus every cycle in T becomes transitive in T (cid:48) .

Original: Now consider any 3 nodes which forms a transitive tournament a > T b > T c and involves at least one node and at most two nodes in T
Revised: Now consider any 3 nodes which forms a transitive tournament a > T b > T c and involves at least one node and at most two nodes in T

Original: + i .
Revised: + i .

Original: Then there are only two cases to consider: (1) a ∈ [ n ] \ T + i and { b, c } ∈ T + i or (2) { a, b } ∈ [ n ] \ T + i and c ∈ T + i .
Revised: Then there are only two cases to consider: (1) a ∈ [ n ] \ T + i and { b, c } ∈ T + i or (2) { a, b } ∈ [ n ] \ T + i and c ∈ T + i .

Original: In both these cases, it is easy to verify that the the corresponding tournament in T (cid:48) is either the transitive tournament b > T (cid:48) >
Revised: In both these cases, it is easy to verify that the the corresponding tournament in T (cid:48) is either the transitive tournament b > T (cid:48) >

Original: c > T (cid:48)
Revised: c > T (cid:48)

Original: a or the transitive tournament c > T (cid:48) a > T (cid:48) b .
Revised: a or the transitive tournament c > T (cid:48) a > T (cid:48) b .

Original: As these are the only possibilities, the result follows by noting that T (cid:48)
Revised: As these are the only possibilities, the result follows by noting that T (cid:48)

Original: =
Revised: =

Original: φ T + i ( T ) =
Revised: φ T + i ( T ) =

Original: ⇒ T =
Revised: ⇒ T =

Original: φ T + i
Revised: φ T + i

Original: ( T (cid:48) ) and so T ∈ F ( T (cid:48) ) .
Revised: ( T (cid:48) ) and so T ∈ F ( T (cid:48) ) .

Original: Proof of Theorem
Revised: Proof of Theorem

Original: Proof.
Revised: Proof.

Original: Assume T is locally transitive.
Revised: Assume T is locally transitive.

Original: Then by Theorem 5, it must be in the ﬂip class of some transitive tournament T (cid:48) i.e. T = φ S ( T (cid:48) ) for some S ⊆
Revised: Then by Theorem 5, it must be in the ﬂip class of some transitive tournament T (cid:48) i.e. T = φ S ( T (cid:48) ) for some S ⊆

Original: [ n ] .
Revised: [ n ] .

Original: It is easy to represent a transitive tournament using a rank 2 skew symmetric matrix.
Revised: It is easy to represent a transitive tournament using a rank 2 skew symmetric matrix.

Original: Indeed pick any vector u ∈ R n which is sorted according to the topological ordering of T (cid:48) .
Revised: Indeed pick any vector u ∈ R n which is sorted according to the topological ordering of T (cid:48) .

Original: Let v ∈ R n be the all ones vector.
Revised: Let v ∈ R n be the all ones vector.

Original: Then
Revised: Then

Original: M = uv T − vu T represents T (cid:48) .
Revised: M = uv T − vu T represents T (cid:48) .

Original: Let M =
Revised: Let M =

Original: ( H (cid:48) )
Revised: ( H (cid:48) )

Original: T A rot H for some H (cid:48)
Revised: T A rot H for some H (cid:48)

Original: ∈ R 2 × n .
Revised: ∈ R 2 × n .

Original: Then the columns of
Revised: Then the columns of

Original: H (cid:48) represent T (cid:48) .
Revised: H (cid:48) represent T (cid:48) .

Original: Now consider the representation H (cid:48) obtained from H (cid:48) where the columns indexed by S are multiplied by − 1 .
Revised: Now consider the representation H (cid:48) obtained from H (cid:48) where the columns indexed by S are multiplied by − 1 .

Original: This does not change the rank of H (cid:48) and it can be veriﬁed that T
Revised: This does not change the rank of H (cid:48) and it can be veriﬁed that T

Original: = T [ H ] .
Revised: = T [ H ] .

Original: To prove the other direction, consider any rank 2 skew symmetric matrix M ∈ R n × n .
Revised: To prove the other direction, consider any rank 2 skew symmetric matrix M ∈ R n × n .

Original: Then there must exist u , v ∈ R n such that M = uv T − vu T .
Revised: Then there must exist u , v ∈ R n such that M = uv T − vu T .

Original: Consider any node i ∈ [ n ] .
Revised: Consider any node i ∈ [ n ] .

Original: Consider three nodes a, b, c such that u i v j > u j v i for j ∈ { a, b, c } .
Revised: Consider three nodes a, b, c such that u i v j > u j v i for j ∈ { a, b, c } .

Original: Furthermore let u a v b > v a u b and u b v c > v b u c .
Revised: Furthermore let u a v b > v a u b and u b v c > v b u c .

Original: Then by carefully going over all {±} sign possibilities for { u i , u a , u b , u c , v i , v a , v b , v c } , one can conclude that it must be the case that u
Revised: Then by carefully going over all {±} sign possibilities for { u i , u a , u b , u c , v i , v a , v b , v c } , one can conclude that it must be the case that u

Original: a v c > v a u c .
Revised: a v c > v a u c .

Original: This just shows that T
Revised: This just shows that T

Original: + i is transitive.
Revised: + i is transitive.

Original: Analogously one can show that T − i is also transitive.
Revised: Analogously one can show that T − i is also transitive.

Original: As i was arbitrary, the result follows.
Revised: As i was arbitrary, the result follows.

Original: Proof of Theorem
Revised: Proof of Theorem

Original: Proof.
Revised: Proof.

Original: Recall that the classic quick sort algorithm picks a pivot node (say 1 ) and places all nodes that beat the pivot to the right in the ranking and those that lose to the left and then recurses on the left and right subsets.
Revised: Recall that the classic quick sort algorithm picks a pivot node (say 1 ) and places all nodes that beat the pivot to the right in the ranking and those that lose to the left and then recurses on the left and right subsets.

Original: As T is locally transitive, choosing any pivot i would correspond to ﬁxing the pivots position and simply returning the ranking [ σ ( N − i ) , i, σ ( N + i )] where σ ( N + i ) , σ ( N − i ) correspond to the topological ordering of the transitive tournaments T ( N + i ) and T ( N − i ) respectively.
Revised: As T is locally transitive, choosing any pivot i would correspond to ﬁxing the pivots position and simply returning the ranking [ σ ( N − i ) , i, σ ( N + i )] where σ ( N + i ) , σ ( N − i ) correspond to the topological ordering of the transitive tournaments T ( N + i ) and T ( N − i ) respectively.

Original: We ﬁrst argue that changing the pivot only cyclically shifts the ﬁnal ranking.
Revised: We ﬁrst argue that changing the pivot only cyclically shifts the ﬁnal ranking.

Original: To see why this is true, consider two pivots i and j and their corresponding rankings σ i and σ j .
Revised: To see why this is true, consider two pivots i and j and their corresponding rankings σ i and σ j .

Original: Without loss of generality, assume that the ranking σ i =
Revised: Without loss of generality, assume that the ranking σ i =

Original: [1 , . . . , n ] and i > T j .
Revised: [1 , . . . , n ] and i > T j .

Original: We argue that there exists an integer k such that N + j = { j +mod n, ( j + 2) mod n, . . .
Revised: We argue that there exists an integer k such that N + j = { j +mod n, ( j + 2) mod n, . . .

Original: , ( j + k ) mod n } (where by convention n mod n = n ).
Revised: , ( j + k ) mod n } (where by convention n mod n = n ).

Original: As N + i is transitive, and j is part of it, it must be the case that all the nodes { j + 1 , . . .
Revised: As N + i is transitive, and j is part of it, it must be the case that all the nodes { j + 1 , . . .

Original: j + n } ∈ N + j .
Revised: j + n } ∈ N + j .

Original: Then to prove the claim, it remains to be shown that the set N − i ∩ N + j is either empty or must be the nodes { 1 , . . .
Revised: Then to prove the claim, it remains to be shown that the set N − i ∩ N + j is either empty or must be the nodes { 1 , . . .

Original: , (cid:96) } for some (cid:96) <
Revised: , (cid:96) } for some (cid:96) <

Original: i .
Revised: i .

Original: If empty, we are done.
Revised: If empty, we are done.

Original: If not, assume for the sake of contradiction that there exists three succesive integers (cid:96) a , (cid:96) b , (cid:96) c < i such that (cid:96) a , (cid:96) c < T j but (cid:96) b > T j .
Revised: If not, assume for the sake of contradiction that there exists three succesive integers (cid:96) a , (cid:96) b , (cid:96) c < i such that (cid:96) a , (cid:96) c < T j but (cid:96) b > T j .

Original: It is easy to verify that this cannot happen as it would lead to T ( { (cid:96) a , (cid:96) b , (cid:96) c , j } ) being a forbidden conﬁguration.
Revised: It is easy to verify that this cannot happen as it would lead to T ( { (cid:96) a , (cid:96) b , (cid:96) c , j } ) being a forbidden conﬁguration.

Original: Notice that as every locally transitive tournament is in the ﬂip class of a transitive tournament i.e.,
Revised: Notice that as every locally transitive tournament is in the ﬂip class of a transitive tournament i.e.,

Original: T = φ S ( T ) for some S , one can divide the set of all nodes into 2 k + 1 groups as follows: Let [1 , . . .
Revised: T = φ S ( T ) for some S , one can divide the set of all nodes into 2 k + 1 groups as follows: Let [1 , . . .

Original: , n ] be the ordering corresponding to the transitive tournament wlog.
Revised: , n ] be the ordering corresponding to the transitive tournament wlog.

Original: Starting from 1 , add as many nodes to a group such that all the elements belong to either S or ¯ S .
Revised: Starting from 1 , add as many nodes to a group such that all the elements belong to either S or ¯ S .

Original: Once the condition is violated, create a new group and continue the same process.
Revised: Once the condition is violated, create a new group and continue the same process.

Original: It is not hard to verify that 2 k + 1 groups will be formed in this process for some k ≥ 0 .
Revised: It is not hard to verify that 2 k + 1 groups will be formed in this process for some k ≥ 0 .

Original: Moreover, each group would separate two other groups by construction.
Revised: Moreover, each group would separate two other groups by construction.

Original: We can ﬁrst show that the items of a single group must appear in consecutive positions in one of the optimal rankings.
Revised: We can ﬁrst show that the items of a single group must appear in consecutive positions in one of the optimal rankings.

Original: This is proven as follows.
Revised: This is proven as follows.

Original: Consider there exists an optimal ranking with items which belong to the same group not occurring consecutively.
Revised: Consider there exists an optimal ranking with items which belong to the same group not occurring consecutively.

Original: Consider two items belonging to the same group, which have items from other groups present in between them in the ranking.
Revised: Consider two items belonging to the same group, which have items from other groups present in between them in the ranking.

Original: Consider these items to be a 1 , a 2 , with a 1 present above in the rankings.
Revised: Consider these items to be a 1 , a 2 , with a 1 present above in the rankings.

Original: Consider the number of upsets that the two items are involved in to be u 1 and u 2 .
Revised: Consider the number of upsets that the two items are involved in to be u 1 and u 2 .

Original: If u 1 ≤ u 2 , a 2 can be placed right after a 1 in the ranking, creating a better or equivalent ranking in terms of upsets.
Revised: If u 1 ≤ u 2 , a 2 can be placed right after a 1 in the ranking, creating a better or equivalent ranking in terms of upsets.

Original: Similarly if u 1 ≥ u 2 , a 1 can be placed directly above a 2 in the rankings to create an equivalent or better ranking.
Revised: Similarly if u 1 ≥ u 2 , a 1 can be placed directly above a 2 in the rankings to create an equivalent or better ranking.

Original: Therefore there exists an optimal ranking which has all items in the same group consecutively.
Revised: Therefore there exists an optimal ranking which has all items in the same group consecutively.

Original: This theorem is then reduced to ﬁnding a ranking of groups, which is proven using induction on k .
Revised: This theorem is then reduced to ﬁnding a ranking of groups, which is proven using induction on k .

Original: Base Case
Revised: Base Case

Original: Consider the base case with k = 1 .
Revised: Consider the base case with k = 1 .

Original: Let there be 3 groups, C , A 1 , B 1 .
Revised: Let there be 3 groups, C , A 1 , B 1 .

Original: We can say that the optimal ranking cannot be any of the following
Revised: We can say that the optimal ranking cannot be any of the following

Original: [Equation]
Revised: [Equation]

Original: since all three rankings can be made better by swapping the second and third ranked groups.
Revised: since all three rankings can be made better by swapping the second and third ranked groups.

Original: Therefore the 3 possible optimal rankings are
Revised: Therefore the 3 possible optimal rankings are

Original: [Equation]
Revised: [Equation]

Original: which are cyclic shifts of each other.
Revised: which are cyclic shifts of each other.

Original: Inductive Step
Revised: Inductive Step

Original: One property of rankings which is useful for the inductive step proof is as follows.
Revised: One property of rankings which is useful for the inductive step proof is as follows.

Original: Let there be 2 k + groups G = { g 1 , g 2 . . .
Revised: Let there be 2 k + groups G = { g 1 , g 2 . . .

Original: g 2 k +1 } .
Revised: g 2 k +1 } .

Original: Label the optimal ranking with the condition that g i be placed ﬁrst in the ranking as R i .
Revised: Label the optimal ranking with the condition that g i be placed ﬁrst in the ranking as R i .

Original: The ranking R i with g i removed must be the optimal ranking for G \ { g i } .
Revised: The ranking R i with g i removed must be the optimal ranking for G \ { g i } .

Original: This can be shown using contradiction i.e, if there was a better ranking for G \ g i , that ranking with g i appended to the front would be better than R i .
Revised: This can be shown using contradiction i.e, if there was a better ranking for G \ g i , that ranking with g i appended to the front would be better than R i .

Original: We now assume the theorem is true for size 2 k − 1 instances and aim to prove for the same for size 2 k + 1 instances.
Revised: We now assume the theorem is true for size 2 k − 1 instances and aim to prove for the same for size 2 k + 1 instances.

Original: Consider g 1 as the ﬁrst group in the ranking.
Revised: Consider g 1 as the ﬁrst group in the ranking.

Original: This creates a certain number of upsets, for the purposes of ranking the remaining groups, 2 of the remaining groups can be merged into a single group.
Revised: This creates a certain number of upsets, for the purposes of ranking the remaining groups, 2 of the remaining groups can be merged into a single group.

Original: This follows from the observation earlier that each group also ’separates’ two groups.
Revised: This follows from the observation earlier that each group also ’separates’ two groups.

Original: This can be considered an instance of the size 2 k − 1 problem.
Revised: This can be considered an instance of the size 2 k − 1 problem.

Original: Therefore the set of optimal rankings with g 1 as the ﬁrst group in the rankings is made up of g 1 as the ﬁrst group and a cyclic sweep of the remaining items to ﬁll the remaining positions.
Revised: Therefore the set of optimal rankings with g 1 as the ﬁrst group in the rankings is made up of g 1 as the ﬁrst group and a cyclic sweep of the remaining items to ﬁll the remaining positions.

Original: Therefore the optimal permutation must be among the sets created by considering each of the 2 k + 1 groups as the ﬁrst group in the rankings.
Revised: Therefore the optimal permutation must be among the sets created by considering each of the 2 k + 1 groups as the ﬁrst group in the rankings.

Original: Let R i,j represent the ranking which has group g i as the ﬁrst group and the remaining groups present as a cyclic sweep from g j .
Revised: Let R i,j represent the ranking which has group g i as the ﬁrst group and the remaining groups present as a cyclic sweep from g j .

Original: Consider the case of R 1 ,k .
Revised: Consider the case of R 1 ,k .

Original: Let x i represent the number of items in group g i .
Revised: Let x i represent the number of items in group g i .

Original: If R 1 ,k is a better ranking than R k,k +1 , it implies that
Revised: If R 1 ,k is a better ranking than R k,k +1 , it implies that

Original: [Equation]
Revised: [Equation]

Original: by considering the shift of g 1 in the two rankings.
Revised: by considering the shift of g 1 in the two rankings.

Original: The difference in the number of upsets between
Revised: The difference in the number of upsets between

Original: R 1 , 2 and R 1 ,k is given by
Revised: R 1 , 2 and R 1 ,k is given by

Original: [Equation]
Revised: [Equation]

Original: Using Equation 1, it can be seen that each of the above terms are negative for any j ≤ n + 1 , making
Revised: Using Equation 1, it can be seen that each of the above terms are negative for any j ≤ n + 1 , making

Original: R 1 , 2 the better ranking.
Revised: R 1 , 2 the better ranking.

Original: Any j > n + 1 cannot be considered as an optimal ranking since the ﬁrst group as per the ranking must precede the second(otherwise switching them would decrease the upsets).
Revised: Any j > n + 1 cannot be considered as an optimal ranking since the ﬁrst group as per the ranking must precede the second(otherwise switching them would decrease the upsets).

Original: Since either R 1 , 2 or R k,k +1 (both counterclockwise orderings) is better than R 1 ,k whenever k ≤ n + 1 ( R 1 ,k cannot be the optimal ranking when k > n + 1 ), and since this can be generalised for any R i,j , it is shown that one of the counterclockwise orderings of the items is the optimal ranking.
Revised: Since either R 1 , 2 or R k,k +1 (both counterclockwise orderings) is better than R 1 ,k whenever k ≤ n + 1 ( R 1 ,k cannot be the optimal ranking when k > n + 1 ), and since this can be generalised for any R i,j , it is shown that one of the counterclockwise orderings of the items is the optimal ranking.

Original: [Equation]
Revised: A.0. F INDING F ORBIDDEN C ONFIGURATIONS

Original: Given a tournament and a representation dimension, there is no known method to check whether the tournament can be represented by vectors in the given dimension.
Revised: Given a tournament and a representation dimension, there is no known method to check whether the tournament can be represented by vectors in the given dimension.

Original: The forbidden conﬁgurations presented above were found by carefully creating an exhaustive set of cases, and showing each one causes a contradiction.
Revised: The forbidden conﬁgurations presented above were found by carefully creating an exhaustive set of cases, and showing each one causes a contradiction.

Original: The techniques used are presented below.
Revised: The techniques used are presented below.

Original: Proof of Theorem
Revised: Proof of Theorem

Original:  Consider the representation of tournaments as given in Subsection 3.
Revised: Proof. Consider the representation of tournaments as given in Subsection 3.

Original: Since adding minor noise to each h i will not change tournament, we can deal only with cases in which any size 4 subset of h 1 , h 2 , ...
Revised: Since adding minor noise to each h i will not change tournament, we can deal only with cases in which any size 4 subset of h 1 , h 2 , ...

Original: h 1 2 consists of linearly independent vectors.
Revised: h 1 2 consists of linearly independent vectors.

Original: By using this property, and without loss of generality,
Revised: By using this property, and without loss of generality,

Original: [Equation]
Revised: [Equation]

Original: We can construct cases based on the signs of the coefﬁcients( c i ) in the above equation.
Revised: We can construct cases based on the signs of the coefﬁcients( c i ) in the above equation.

Original: There are 2 4 = 16 sign patterns/cases for any equation.
Revised: There are 2 4 = 16 sign patterns/cases for any equation.

Original: These cases are ﬁltered by multiplying the entire expression with expressions of the form A rot h i .
Revised: These cases are ﬁltered by multiplying the entire expression with expressions of the form A rot h i .

Original: [Equation]
Revised: [Equation]

Original: In the above equation, the signs of all expressions of the form h
Revised: In the above equation, the signs of all expressions of the form h

Original: i A rot h j is known from the tournament conﬁguration.
Revised: i A rot h j is known from the tournament conﬁguration.

Original: Therefore simply comparing the sign of the LHS and RHS for all possible values of i rules out many cases.
Revised: Therefore simply comparing the sign of the LHS and RHS for all possible values of i rules out many cases.

Original: We now use multiple equations together in a bid to further ﬁlter the remaining cases.
Revised: We now use multiple equations together in a bid to further ﬁlter the remaining cases.

Original: Consider the following equations without loss of generality.
Revised: Consider the following equations without loss of generality.

Original: [Equation]
Revised: [Equation]

Original: h 5 can be eliminated from the ﬁrst two equations, leaving two expressions of h 6 in terms of h 1 , . . .
Revised: h 5 can be eliminated from the ﬁrst two equations, leaving two expressions of h 6 in terms of h 1 , . . .

Original: h 4 .
Revised: h 4 .

Original: The two sets of coefﬁcients of h 1 , . . .
Revised: The two sets of coefﬁcients of h 1 , . . .

Original: h 4 can be equated, and sign based arguments can eliminate a few more cases.
Revised: h 4 can be equated, and sign based arguments can eliminate a few more cases.

Original: Also, a set of 3 tuples can be constructed, with each item representing possible sign patterns for the 3 equations.
Revised: Also, a set of 3 tuples can be constructed, with each item representing possible sign patterns for the 3 equations.

Original: Note that this set of 3 tuples does not contain entries which cannot apply simultaneously on the 3 equations.
Revised: Note that this set of 3 tuples does not contain entries which cannot apply simultaneously on the 3 equations.

Original: The above elimination step can be considered as a ﬁltration procedure given a size 6 tuple ( h 1 , h 2 , h 3 , h 4 , h 5 , h 6 ) by using 3 equations.
Revised: The above elimination step can be considered as a ﬁltration procedure given a size 6 tuple ( h 1 , h 2 , h 3 , h 4 , h 5 , h 6 ) by using 3 equations.

Original: We can perform a similar ﬁltration step given any size 6 tuple as well.
Revised: We can perform a similar ﬁltration step given any size 6 tuple as well.

Original: Let the equations corresponding to ( h 1 , h 2 , h 3 , h 4 , h 5 , h 6 ) be E 1 , 1 , E 1 , 2 , E 1 , 3 .
Revised: Let the equations corresponding to ( h 1 , h 2 , h 3 , h 4 , h 5 , h 6 ) be E 1 , 1 , E 1 , 2 , E 1 , 3 .

Original: Similarly, consider the tuples ( h 1 , h 2 , h 3 , h 4 , h 5 , h 7 ) , ( h 2 , h 3 , h 4 , h 5 , h 6 , h 7 ) , ( h 1 , h 2 , h 3 , h 4 , h 6 , h 7 ) and their corresponding equations represented by E i,j where i is the index of the tuple it corresponds to and j is the index of the equation.
Revised: Similarly, consider the tuples ( h 1 , h 2 , h 3 , h 4 , h 5 , h 7 ) , ( h 2 , h 3 , h 4 , h 5 , h 6 , h 7 ) , ( h 1 , h 2 , h 3 , h 4 , h 6 , h 7 ) and their corresponding equations represented by E i,j where i is the index of the tuple it corresponds to and j is the index of the equation.

Original: The above ﬁltration process can be performed on all the tuples, following which an additional ﬁltering step can be performed by using
Revised: The above ﬁltration process can be performed on all the tuples, following which an additional ﬁltering step can be performed by using

Original: • E 1 , 1 ≡ E 2 ,• E 1 , 2 ≡ E 3 ,• E 1 , 3 ≡ E 4 ,• E 2 , 2 ≡ E 3 ,• E 2 , 3 ≡ E 4 ,
Revised: • E 1 , 1 ≡ E 2 ,• E 1 , 2 ≡ E 3 ,• E 1 , 3 ≡ E 4 ,• E 2 , 2 ≡ E 3 ,• E 2 , 3 ≡ E 4 ,

Original: where the equivalence relation represents that the 2 equations are identical.
Revised: where the equivalence relation represents that the 2 equations are identical.

Original: Since identical equations must have identical sign patterns, the sign patterns not present in both sets of possible sign patterns can be ﬁltered out.
Revised: Since identical equations must have identical sign patterns, the sign patterns not present in both sets of possible sign patterns can be ﬁltered out.

Original: For the 11-DRT cone, this leaves us with null set, proving that it is a forbidden conﬁguration. 
Revised: For the 11-DRT cone, this leaves us with null set, proving that it is a forbidden conﬁguration. Proof of Theorem Proof.

Revised: The result follows the arguments in Forster & Simon (2006).

Original: A.0. A DDITIONAL F ILTRATION FOR VERIFYING 10 N ODE F ORBIDDEN C ONFIGURATION
Revised: We note that the arguments in Forster & Simon (2006) work only for non-zero sign matrices. By overloading T to denote the signed adjacency matrix of the corresponding tournament, we can consider the non-zero sign matrix G = T + diag ( b ) where b ∈ { 1 , − 1 } n . By Gershgorin’s circle theorem and exploting the fact that any two rows of a doubly regular tournament are orthogonal, one can show that ρ i ( G ) 2 ≤ ρ i ( T ) 2 + 2 n − 2 where ρ i denotes the i -th largest singular value of the corresponding matrix. It then follows from Forster & Simon (2006) that if G has an representation in dim dimensions, then it must be the case that

Revised: [Equation]

Revised: Noting that for a doubly regular cone T on n nodes, ρ i ( T ) .

Revised: 2 = n − 1 ∀

Revised: i , we get

Revised: [Equation]

Revised: Thus to get a matrix M such that sign ( M ) =

Revised: G , one needs at least (cid:112) n 3 dimensions i.e., rank ( M ) ≥ (cid:112) n 3 .

Revised: Now we show that this also is a lower bound for representing T .

Revised: To see this, assume for the sake of contradiction that T has a representation in d dimension where d < (cid:112) n 3 .

Revised: Then, there exists H ∈ R d × n such that T

Revised: = T [ H ] .

Revised: The diagonal entries of T [ H ] are zero.

Revised: Now introduce a small enough perturbation E to H and consider the matrix ( H + E ) T A rot H .

Revised: The entries of the perturbation matrix E ∈ R d × n can be chosen to be small enough such that the sign of the off-diagonal entries of H T A rot H is same as that of ( H + E ) T A rot H .

Revised: However, the diagonal entries of ( H + E ) T A rot H can get an arbitrary sign pattern, say b .

Revised: Let G be the sign pattern matrix corresponding to ( H + E ) T A rot H .

Revised: By deﬁnition, G has a representation in dimension d < (cid:112) n 3 .

Revised: But this contradicts inequality (7).

Revised: Thus, it must be the case that T has a representation in dimension at least (cid:112) n 3 .

Revised: Finally, setting n = 12 d 2 to the number of nodes in the doubly regular cone as given in the Theorem, we get that at least 2 d dimensions are needed to embed such a tournament.

Revised: The result follows.

Revised: Proof of Theorem

Revised: Proof.

Revised: The proof is the same arguments as the ﬁrst part of the proof of Theorem 9.

Revised: Proof of Theorem

Revised: Proof.

Revised: Given a tournament T , we ﬁrst show that we can start with an arbitrary transitive tournament and add enough rank 2 corrections to obtain a representation for T .

Revised: Every addition of a rank 2 matrix will increase the representation dimension by at most 2 .

Revised: The result will follow then by noting that for the choice of the transitive tournament which minimizes the number of corrections needed, one needs at most 2( µ ( T ) + 1) representation dimension.

Revised: Let T (cid:48) be an arbitrary transitive tournament which has a 2 dimensional representation and let

Revised: M ∈ R n × n be the associated skew symmetric matrix that represents T (cid:48) .

Revised: W.l.o.g, assume that

Revised: M ij > 0 if and only if i < j .

Revised: Deﬁne E k ( T )

Revised: =

Revised: { i : i < k, i < T k } .

Revised: By deﬁnition, the feedback arc set E σ ( T ) =

Revised: ∪ nk =1 ( k, E k ( T ))

Revised: where σ = [1 , . . .

Revised: , n ] is the topological order corresponding to the transitive tournament T (cid:48) .

Revised: We start with k = n and correct the feedback arc errors arising from E k iteratively.

Revised: Deﬁne ∆ n = (cid:113) max i<n ( M in + (cid:15) ) for some small enough (cid:15) > 0 .

Revised: Let u ∈ R n be such that u i =

Original: {+∆ ∀ i ∈ E n and u i = 
+}
Revised: {+∆ ∀ i ∈ E n and u i = 
+}

Revised: otherwise.

Revised: Let v ∈ R n be such that v n = − ∆ , v i = 0 ∀ i (cid:54) = n .

Original: {+It is easy to verify that M + uv T − vu T represents a tournament that has the feedback arc set ∪ n − 1 k =
+}
Revised: {+It is easy to verify that M + uv T − vu T represents a tournament that has the feedback arc set ∪ n − 1 k =
+}

Revised: ( k, E k ( T ))

Revised: i.e., the errors in E n have been corrected.

Revised: The cost of correcting the error is adding a rank 2 skew symmetric matrix which increases the representation dimension by at most 2 .

Revised: One can repeat the same procedure for n − 1 , n − 2 , . . .

Revised: until all errors are corrected.

Revised: The upper bound in the theorem follows noting that the above argument works for any transitive tournament T (cid:48) and so we can start with the one which has the least number of nodes involved in the feedback arc set to minimize the number of extra dimensions needed to represent T .

Revised: Proof of Theorem

Revised: Proof.

Revised: From Theorem 11, T can be represented using at most 2( µ ( T ) + 1) dimensions.

Revised: By construction any representation of T must also represent G .

Revised: The result follows.

Revised: A.0. A DDITIONAL F ILTRATION FOR VERIFYING 10 N ODE F ORBIDDEN C ONFIGURATION

Original: T =     1 − 1 − 1 − 1 − 1 − − 1 − 1 − 1 −1 −− 1 −  1 −1 − 1 − 1 −−1 −0 − 1 − 1 − 1 −1 − 1 −  1 − 1 −1 −1 −1 − 1 − 1 −1 − 1 − 1 −1 −0 −1 −        1 −1 − 1 − 1 − 1 − 1 −0 −−          
Revised: T =     1 − 1 − 1 − 1 − 1 − − 1 − 1 − 1 −1 −− 1 −  1 −1 − 1 − 1 −−1 −0 − 1 − 1 − 1 −1 − 1 −  1 − 1 −1 −1 −1 − 1 − 1 −1 − 1 − 1 −1 −0 −1 −        1 −1 − 1 − 1 − 1 − 1 −0 −−          

Original: Due to the lesser number of constraints present when dealing with 10 nodes instead of 12, an additional ﬁltration step is required.
Revised: Due to the lesser number of constraints present when dealing with 10 nodes instead of 12, an additional ﬁltration step is required.

Original: The entire procedure described above can be considered as an elimination procedure given a tuple T 1 = ( h 1 , . . .
Revised: The entire procedure described above can be considered as an elimination procedure given a tuple T 1 = ( h 1 , . . .

Original: h 7 ) .
Revised: h 7 ) .

Original: Also consider that in the ﬁnal ﬁltration above, a set of size 4 tuples is created.
Revised: Also consider that in the ﬁnal ﬁltration above, a set of size 4 tuples is created.

Original: These size 4 tuples will represent the possible sign patterns for E 1 , 1 , E 1 , 2 , E 1 , 3 , E 2 , 2 .
Revised: These size 4 tuples will represent the possible sign patterns for E 1 , 1 , E 1 , 2 , E 1 , 3 , E 2 , 2 .

Original: If a similar procedure is carried out on T 2 = ( h 2 , . . .
Revised: If a similar procedure is carried out on T 2 = ( h 2 , . . .

Original: h 8 ) , another set oftuples will be generated.
Revised: h 8 ) , another set oftuples will be generated.

Original: The ﬁltration in this step is based on the fact that the equations E 1 , 3 , E 2 ,corresponding to T 1 are the same as E 1 , 1 , E 1 , 2 for T 2 .
Revised: The ﬁltration in this step is based on the fact that the equations E 1 , 3 , E 2 ,corresponding to T 1 are the same as E 1 , 1 , E 1 , 2 for T 2 .

Original: The sign patterns which are not present in both sets of tuples are ﬁltered out.
Revised: The sign patterns which are not present in both sets of tuples are ﬁltered out.

Original: This type of relationship can be obtained between T 2 and
Revised: This type of relationship can be obtained between T 2 and

Original: T 3 =
Revised: T 3 =

Original: ( h 3 , . . .
Revised: ( h 3 , . . .

Original: h 8 , h 0 ) as well, and in general between any T i and T j such that T j can be obtained from T i by removing the ﬁrst entry and adding an entry to the end.
Revised: h 8 , h 0 ) as well, and in general between any T i and T j such that T j can be obtained from T i by removing the ﬁrst entry and adding an entry to the end.

Original: Note that the tuples must have unique entries.
Revised: Note that the tuples must have unique entries.

Original: Therefore a ’circular elimination’ procedure can be followed, where the T i tuples considered are size 7 subsets of the circular permutations of h 1 , . . .
Revised: Therefore a ’circular elimination’ procedure can be followed, where the T i tuples considered are size 7 subsets of the circular permutations of h 1 , . . .

Original: h 8 .
Revised: h 8 .

Original: By following this procedure, all the sign patterns remaining can be eliminated.
Revised: By following this procedure, all the sign patterns remaining can be eliminated.

Original: Using size 7 circular permutations of h 1 , . . .
Revised: Using size 7 circular permutations of h 1 , . . .

Original: h 1 0 in the last step does not eliminate all the cases, we believe this is due to nodes 1 to 8 forming a 7-DRT cone, which leads to many eliminations.
Revised: h 1 0 in the last step does not eliminate all the cases, we believe this is due to nodes 1 to 8 forming a 7-DRT cone, which leads to many eliminations.

Original: The anonymized code 
Revised: The anonymized code for the ﬁltration can be found here .