Original: Faster and Scalable Algorithms for Densest Subgraph and Decomposition
Revised: Faster and Scalable Algorithms for Densest Subgraph and Decomposition

Original: Anonymous Author(s)
Revised: Anonymous Author(s)

Original: AfﬁliationAddress email
Revised: AfﬁliationAddress email

Original: Abstract
Revised: Abstract

Original: We study the densest subgraph problem (DSG) and the densest subgraph localdecomposition problem (DSG-LD) in undirected graphs.
Revised: We study the densest subgraph problem (DSG) and the densest subgraph localdecomposition problem (DSG-LD) in undirected graphs.

Original: We also consider supermodular generalizations of these problems.
Revised: We also consider supermodular generalizations of these problems.

Original: For large scale graphs simple iterative algorithms perform much better in practice than theoretically fast algorithmsbased on network-ﬂow or LP solvers.
Revised: For large scale graphs simple iterative algorithms perform much better in practice than theoretically fast algorithmsbased on network-ﬂow or LP solvers.

Original: Boob et al.
Revised: Boob et al.

Original: [1] recently gave a fast iterativealgorithm called G REEDY ++ for DSG.
Revised: [1] recently gave a fast iterativealgorithm called G REEDY ++ for DSG.

Original: It was shown in [2] that it converges toa (1  ✏ ) relative approximation to the optimum density in O ( 1 ✏ 2  ( G )  ⇤ ) iterationswhere
Revised: It was shown in [2] that it converges toa (1  ✏ ) relative approximation to the optimum density in O ( 1 ✏ 2  ( G )  ⇤ ) iterationswhere

Original:  ( G ) is the maximum degree and  ⇤ is the optimum density.
Revised:  ( G ) is the maximum degree and  ⇤ is the optimum density.

Original: Danisch et al.[3] gave an iterative algorithm based on the Frank-Wolfe algorithm for DSG-LDthat takes O ( m  ( G ) ✏) iterations to converge to an ✏ -additive approximate local decomposition vector ˆ b , where m is number of edges in the graph.
Revised: Danisch et al.[3] gave an iterative algorithm based on the Frank-Wolfe algorithm for DSG-LDthat takes O ( m  ( G ) ✏) iterations to converge to an ✏ -additive approximate local decomposition vector ˆ b , where m is number of edges in the graph.

Original: In this paper we give a new iterative algorithm for both problems that takes at most
Revised: In this paper we give a new iterative algorithm for both problems that takes at most

Original: O ( p m  ( G ) ✏ ) iterations to converge to an ✏ -additive approximate local decomposition vector; each iteration can be implemented in O ( m ) time.
Revised: O ( p m  ( G ) ✏ ) iterations to converge to an ✏ -additive approximate local decomposition vector; each iteration can be implemented in O ( m ) time.

Original: We describe afractional peeling technique which has strong empirical performance as well astheoretical guarantees.
Revised: We describe afractional peeling technique which has strong empirical performance as well astheoretical guarantees.

Original: The algorithm is scalable and simple, and can be appliedto graphs with hundreds of millions of edges.
Revised: The algorithm is scalable and simple, and can be appliedto graphs with hundreds of millions of edges.

Original: We test our algorithm on real andsynthetic data sets and show that it provides a signiﬁcant beneﬁt over previousalgorithms.
Revised: We test our algorithm on real andsynthetic data sets and show that it provides a signiﬁcant beneﬁt over previousalgorithms.

Original: The algorithm and analysis extends to hypergraphs.
Revised: The algorithm and analysis extends to hypergraphs.

Original: 1. Introduction
Revised: 1. Introduction

Original: The densest subgraph problem (DSG) is a classical problem in combinatorial optimization and hasmany real world applications in data mining, network analysis and machine learning.
Revised: The densest subgraph problem (DSG) is a classical problem in combinatorial optimization and hasmany real world applications in data mining, network analysis and machine learning.

Original: The input for
Revised: The input for

Original: DSG is an undirected graph G =
Revised: DSG is an undirected graph G =

Original: ( V, E ) with m = | E | and n = | V | .
Revised: ( V, E ) with m = | E | and n = | V | .

Original: The goal is to return a subset
Revised: The goal is to return a subset

Original: S ✓ V that maximizes | E ( S ) | | S | where E ( S ) =
Revised: S ✓ V that maximizes | E ( S ) | | S | where E ( S ) =

Original: {{ u, v } 2 E : u, v 2 S } is the set of edges withboth end points in S .
Revised: {{ u, v } 2 E : u, v 2 S } is the set of edges withboth end points in S .

Original: DSG has a variety of applications in which dense subgraphs reveal importantinformation about the underlying network such as communities.
Revised: DSG has a variety of applications in which dense subgraphs reveal importantinformation about the underlying network such as communities.

Original: One can view it as a subroutine inan unsupervised clustering procedure.
Revised: One can view it as a subroutine inan unsupervised clustering procedure.

Original: It is a canonical problem in the broad area of dense subgraphdiscovery which has seen many developments and applications in the past two decades.
Revised: It is a canonical problem in the broad area of dense subgraphdiscovery which has seen many developments and applications in the past two decades.

Original: We point thereader’s attention to a (non-exhaustive) list of recent, and some not so recent, important work and thepointers therein [4, 5, 6, 7, 1, 3, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].
Revised: We point thereader’s attention to a (non-exhaustive) list of recent, and some not so recent, important work and thepointers therein [4, 5, 6, 7, 1, 3, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].

Original: Constrainedversions of DSG such as the densest k -subgraph problem D K SG, densest at most k -subgraph, anddensest at least k -subgraph D ALK SG are also well-studied and many of these are NP-Hard.
Revised: Constrainedversions of DSG such as the densest k -subgraph problem D K SG, densest at most k -subgraph, anddensest at least k -subgraph D ALK SG are also well-studied and many of these are NP-Hard.

Original: See[24, 25, 26, 10, 6, 27] for some positive and negative results on approximation algorithms.
Revised: See[24, 25, 26, 10, 6, 27] for some positive and negative results on approximation algorithms.

Original: One of the key advantages of DSG is its polynomial-time solvability.
Revised: One of the key advantages of DSG is its polynomial-time solvability.

Original: The ﬁrst polynomial-timealgorithms were due to Goldberg [4] and Picard and Queuranne [28].
Revised: The ﬁrst polynomial-timealgorithms were due to Goldberg [4] and Picard and Queuranne [28].

Original: The decision version of DSG is the following: given G and a rational number  , is the density in G at least  ?
Revised: The decision version of DSG is the following: given G and a rational number  , is the density in G at least  ?

Original: The algorithms in[4, 28] construct an auxiliary directed ﬂow-network H such that the maximum ﬂow in H allows oneto answer the decision version.
Revised: The algorithms in[4, 28] construct an auxiliary directed ﬂow-network H such that the maximum ﬂow in H allows oneto answer the decision version.

Original: Binary search over  leads to the ﬁnal algorithm.
Revised: Binary search over  leads to the ﬁnal algorithm.

Original: Maximum ﬂowis a powerful algorithmic subroutine, however, it is not a practical or scalable algorithm for moderngraph data sets with millions and even billions of vertices and edges.
Revised: Maximum ﬂowis a powerful algorithmic subroutine, however, it is not a practical or scalable algorithm for moderngraph data sets with millions and even billions of vertices and edges.

Original: For example, authors from [1]noted that the Goldberg’s maximum ﬂow algorithm failed on many large scale graphs even thoughthey used a highly optimized maximum ﬂow library.
Revised: For example, authors from [1]noted that the Goldberg’s maximum ﬂow algorithm failed on many large scale graphs even thoughthey used a highly optimized maximum ﬂow library.

Original: Charikar [5] described a linear programmingformulation for DSG that gives an exact solution and has O ( | E | ) variables and constraints.
Revised: Charikar [5] described a linear programmingformulation for DSG that gives an exact solution and has O ( | E | ) variables and constraints.

Original: LPsolvers are also unsuitable for large data sets due to memory limitations among others.
Revised: LPsolvers are also unsuitable for large data sets due to memory limitations among others.

Original: Charikar [5]described a very simple 12 -approximation algorithm for DSG known as the G REEDY or P EELINGalgorithm.
Revised: Charikar [5]described a very simple 12 -approximation algorithm for DSG known as the G REEDY or P EELINGalgorithm.

Original: The algorithm creates an ordering of the vertices as follows.
Revised: The algorithm creates an ordering of the vertices as follows.

Original: The ﬁrst vertex v 1 is the onewith the smallest degree in G (ties broken arbitrarily).
Revised: The ﬁrst vertex v 1 is the onewith the smallest degree in G (ties broken arbitrarily).

Original: It selects v 2 to be the smallest degree vertexin G  v 1 .
Revised: It selects v 2 to be the smallest degree vertexin G  v 1 .

Original: Letting G i be the graph after removing v 1 , v 2 , . .
Revised: Letting G i be the graph after removing v 1 , v 2 , . .

Original: . , v i  1 (with G 0 = G ), the algorithmreturns the graph among G 0 , . .
Revised: . , v i  1 (with G 0 = G ), the algorithmreturns the graph among G 0 , . .

Original: . , G n with the highest density.
Revised: . , G n with the highest density.

Original: The algorithm can be implementedin O ( m ) time.
Revised: The algorithm can be implementedin O ( m ) time.

Original: The ordering created by the algorithm is the same as the one to compute a k -coredecomposition of a graph — this is a well-studied graph decomposition procedure with severalapplications [29] .
Revised: The ordering created by the algorithm is the same as the one to compute a k -coredecomposition of a graph — this is a well-studied graph decomposition procedure with severalapplications [29] .

Original: The simplicity and the efﬁciency of the Greedy algorithm, and its approximationguarantees, has led to its adoption for a number of other density measures.
Revised: The simplicity and the efﬁciency of the Greedy algorithm, and its approximationguarantees, has led to its adoption for a number of other density measures.

Original: Despite Greedy’s advantages, its worst-case approximation guarantee is only 1 2 , and is worse forother density measures.
Revised: Despite Greedy’s advantages, its worst-case approximation guarantee is only 1 2 , and is worse forother density measures.

Original: The goal is to develop algorithms that obtain a (1  " ) relative approximationfor a given parameter " 2 (0 , 1) while also being scalable to large graphs.
Revised: The goal is to develop algorithms that obtain a (1  " ) relative approximationfor a given parameter " 2 (0 , 1) while also being scalable to large graphs.

Original: One approach to obtainsuch algorithms is via the dual of Charikar’s LP relaxation.
Revised: One approach to obtainsuch algorithms is via the dual of Charikar’s LP relaxation.

Original: It is a mixed packing and covering LPfor a given guess of the optimal value.
Revised: It is a mixed packing and covering LPfor a given guess of the optimal value.

Original: Such LPs can be approximately solved via iterative methodssuch as the multiplicative weight updates (MWU) or other methods based on convex optimization.
Revised: Such LPs can be approximately solved via iterative methodssuch as the multiplicative weight updates (MWU) or other methods based on convex optimization.

Original: Bahmani, Goel and Munagala [30] applied this methodology to obtain an algorithm that yields a(1  " ) -approximation in ˜ O
Revised: Bahmani, Goel and Munagala [30] applied this methodology to obtain an algorithm that yields a(1  " ) -approximation in ˜ O

Original: ( m/ " 2 )
Revised: ( m/ " 2 )

Original: -time.
Revised: -time.

Original: Boob, Sawlani and Wang [31] described an algorithm thatyields a (1  " ) -approximation in ˜ O
Revised: Boob, Sawlani and Wang [31] described an algorithm thatyields a (1  " ) -approximation in ˜ O

Original: ( m  ( G ) / " ) -time, where  ( G ) is the maximum degree in G .
Revised: ( m  ( G ) / " ) -time, where  ( G ) is the maximum degree in G .

Original: More recently Chekuri, Quanrud and Torres [2] obtained a (1  " ) -approximation in ˜ O
Revised: More recently Chekuri, Quanrud and Torres [2] obtained a (1  " ) -approximation in ˜ O

Original: ( m/ " )
Revised: ( m/ " )

Original: -timevia approximate ﬂow techniques.
Revised: -timevia approximate ﬂow techniques.

Original: Some of these nice theoretical developments have not yet led topractically useful algorithms for large scale graphs.
Revised: Some of these nice theoretical developments have not yet led topractically useful algorithms for large scale graphs.

Original: In a different direction, Boob et al.
Revised: In a different direction, Boob et al.

Original: [1] describeda fast iterative algorithm called G REEDY ++ which builds on the Greedy algorithm and insights fromthe LP relaxation.
Revised: [1] describeda fast iterative algorithm called G REEDY ++ which builds on the Greedy algorithm and insights fromthe LP relaxation.

Original: It does extremely well in experiments, and the authors conjectured that it yields a(1  ✏ ) relative approximation in O ( 1 ✏ 2 ) iterations.
Revised: It does extremely well in experiments, and the authors conjectured that it yields a(1  ✏ ) relative approximation in O ( 1 ✏ 2 ) iterations.

Original: (Each iteration takes O ( m ) time.)
Revised: (Each iteration takes O ( m ) time.)

Original: Chekuri et al.[2] proved that G REEDY ++ converges to a (1  ✏ ) -approximation in O (  ( G )  ⇤ ✏ 2 ) iterations where  ⇤is the optimum density.
Revised: Chekuri et al.[2] proved that G REEDY ++ converges to a (1  ✏ ) -approximation in O (  ( G )  ⇤ ✏ 2 ) iterations where  ⇤is the optimum density.

Original: This gives evidence of the theoretical soundness of the algorithm.
Revised: This gives evidence of the theoretical soundness of the algorithm.

Original: Our goal in this paper is to develop new and scalable algorithms that outperform G REEDY +
Revised: Our goal in this paper is to develop new and scalable algorithms that outperform G REEDY +

Original: + whilehaving strong theoretical guarantees.
Revised: + whilehaving strong theoretical guarantees.

Original: In addition, we are interested in an algorithm that ﬁnds anapproximate dense subgraph decomposition which gives information on the density structure of thegraph and is different from another such decomposition, namely the k -core decomposition.
Revised: In addition, we are interested in an algorithm that ﬁnds anapproximate dense subgraph decomposition which gives information on the density structure of thegraph and is different from another such decomposition, namely the k -core decomposition.

Original: Further,as shown in previous work [32, 33, 2], a dense subgraph decomposition allows one to compactly represent approximate solutions to densest subgraphs of different sizes; in particular the representationyields a 2 -approximation for densest at least k -subgraph for any given k [32, 2].
Revised: Further,as shown in previous work [32, 33, 2], a dense subgraph decomposition allows one to compactly represent approximate solutions to densest subgraphs of different sizes; in particular the representationyields a 2 -approximation for densest at least k -subgraph for any given k [32, 2].

Original: Dense subgraph decomposition.
Revised: Dense subgraph decomposition.

Original: One can show that every graph G =
Revised: One can show that every graph G =

Original: ( V, E ) admits a certainstructured nested decomposition based on decreasing density.
Revised: ( V, E ) admits a certainstructured nested decomposition based on decreasing density.

Original: In particular, the vertex set V canpartitioned into S 1 , S 2 , . . .
Revised: In particular, the vertex set V canpartitioned into S 1 , S 2 , . . .

Original: , S k such that S 1 is the unique maximal densest subgraph in G , and S i isthe unique maximal densest set with respect to S 1 [ . . .
Revised: , S k such that S 1 is the unique maximal densest subgraph in G , and S i isthe unique maximal densest set with respect to S 1 [ . . .

Original: [ S i  1 (a formal deﬁnition is deferred to 81 Section 3).
Revised: [ S i  1 (a formal deﬁnition is deferred to 81 Section 3).

Original: For each vertex v we let  v denote the density of the set containing v .
Revised: For each vertex v we let  v denote the density of the set containing v .

Original: The existence ofsuch a decomposition follows from the supermodularity properties of the function f ( S ) =
Revised: The existence ofsuch a decomposition follows from the supermodularity properties of the function f ( S ) =

Original: | E ( S ) | ,and is implicitly known from past work on submodular functions (in particular, the classical resultof Fujishige [34]).
Revised: | E ( S ) | ,and is implicitly known from past work on submodular functions (in particular, the classical resultof Fujishige [34]).

Original: For graphs this decomposition was rediscovered by Tatti and Gionis [35, 12]under the name of locally-dense decomposition.
Revised: For graphs this decomposition was rediscovered by Tatti and Gionis [35, 12]under the name of locally-dense decomposition.

Original: Tatti showed that one can compute the exact decomposition via n maximum ﬂow computations.
Revised: Tatti showed that one can compute the exact decomposition via n maximum ﬂow computations.

Original: Danisch et al.
Revised: Danisch et al.

Original: [3] showed that computing the  vvalues can be cast as solving a quadratic program that extends the LP of Charikar, and applied the
Revised: [3] showed that computing the  vvalues can be cast as solving a quadratic program that extends the LP of Charikar, and applied the

Original: Frank-Wolfe algorithm.
Revised: Frank-Wolfe algorithm.

Original: They showed that their algorithm needs O ( | E |  ( G ) ✏) iterations (each taking O ( | E | ) time) to converge to an " -approximate vector.
Revised: They showed that their algorithm needs O ( | E |  ( G ) ✏) iterations (each taking O ( | E | ) time) to converge to an " -approximate vector.

Original: Their work shows that the Frank-Wolfealgorithm leads to a good approximation to the densest subgraph in a relatively small number of iterations based on experiments.
Revised: Their work shows that the Frank-Wolfealgorithm leads to a good approximation to the densest subgraph in a relatively small number of iterations based on experiments.

Original: However, they do not describe a systematic way to extract a densesubgraph decomposition from the approximate vector with provable guarantees. 
Revised: However, they do not describe a systematic way to extract a densesubgraph decomposition from the approximate vector with provable guarantees. See Figure 6.4 in

Revised: Appendix 6.1 for a table summarizing known results.

Original: Densest Supermodular Set.
Revised: Densest Supermodular Set.

Original: Efﬁcient solvability of DSG can also be seen via a connection to amore general problem called the densest supermodular set problem, which we refer to as DSS.
Revised: Efﬁcient solvability of DSG can also be seen via a connection to amore general problem called the densest supermodular set problem, which we refer to as DSS.

Original: Areal-valued set function f : 2 V !
Revised: Areal-valued set function f : 2 V !

Original: R + is said to be supermodular iff f ( A ) + f ( B )  f ( A [ B ) +f ( A \ B ) for all A, B ✓ V .
Revised: R + is said to be supermodular iff f ( A ) + f ( B )  f ( A [ B ) +f ( A \ B ) for all A, B ✓ V .

Original: The goal in DSS is to return S ✓ V that maximizes f ( S ) | S | .
Revised: The goal in DSS is to return S ✓ V that maximizes f ( S ) | S | .

Original: For anygraph G =
Revised: For anygraph G =

Original: ( V, E ) , the function f : 2 V !
Revised: ( V, E ) , the function f : 2 V !

Original: R + deﬁned as f ( S ) =
Revised: R + deﬁned as f ( S ) =

Original: | E ( S ) | for each S ✓ V , is knownto be supermodular.
Revised: | E ( S ) | for each S ✓ V , is knownto be supermodular.

Original: Hence DSS generalizes DSG.
Revised: Hence DSS generalizes DSG.

Original: Several algorithms and structural features for
Revised: Several algorithms and structural features for

Original: DSG are easier to understand via supermodularity, as shown in recent work [2].
Revised: DSG are easier to understand via supermodularity, as shown in recent work [2].

Original: Notation: For an undirected graph G =
Revised: Notation: For an undirected graph G =

Original: ( V, E ) and S 1 , S 2 ✓ V , we use E ( S 1 , S 2 ) for the edgeset {{ u, v } 2 E : u 2 S 1 , v 2 S 2 } .
Revised: ( V, E ) and S 1 , S 2 ✓ V , we use E ( S 1 , S 2 ) for the edgeset {{ u, v } 2 E : u 2 S 1 , v 2 S 2 } .

Original: For a vector b 2 R V , and a subset S ✓ V , we let b ( S ) 102 denote
Revised: For a vector b 2 R V , and a subset S ✓ V , we let b ( S ) 103 denote

Original: P i 2 S b i .
Revised: P i 2 S b i .

Original: We let ord ( E ) be the set of 2 | E | edges that are ordered edges uv and vu for eachunordered edge { u, v } 2 E .
Revised: We let ord ( E ) be the set of 2 | E | edges that are ordered edges uv and vu for eachunordered edge { u, v } 2 E .

Original: Finally, m and n denote | E | and | V | respectively.
Revised: Finally, m and n denote | E | and | V | respectively.

Original: 2. Technical Contributions
Revised: 2. Technical Contributions

Original: [Figure]

Original: We summarize the main contributions of this paper.
Revised: We summarize the main contributions of this paper.

Original: A fast and scalable algorithm based on projections: We describe an iterative algorithm for DSGand dense subgraph decomposition based on solving a quadratic objective with linear constraintsthat is derived from the dual of Charikar’s LP.
Revised: A fast and scalable algorithm based on projections: We describe an iterative algorithm for DSGand dense subgraph decomposition based on solving a quadratic objective with linear constraintsthat is derived from the dual of Charikar’s LP.

Original: Unlike previous work [36] that relied on the FrankWolfe method, we use a projection-based approach.
Revised: Unlike previous work [36] that relied on the FrankWolfe method, we use a projection-based approach.

Original: The algorithm is extremely simple and highlyparallelizable.
Revised: The algorithm is extremely simple and highlyparallelizable.

Original: We show that it converges to an " -additive approximation in O ( p m  ( G ) / " ) iterations which is signiﬁcantly better than the bound for the Frank-Wolfe method.
Revised: We show that it converges to an " -additive approximation in O ( p m  ( G ) / " ) iterations which is signiﬁcantly better than the bound for the Frank-Wolfe method.

Original: The algorithm scalesto extremely large graphs and outperforms existing algorithms in both running time and the approximation quality on almost all data sets.
Revised: The algorithm scalesto extremely large graphs and outperforms existing algorithms in both running time and the approximation quality on almost all data sets.

Original: The algorithm and analysis generalizes to hypergraphs.
Revised: The algorithm and analysis generalizes to hypergraphs.

Original: Fractional Peeling : We introduce the idea of fractional peeling to round a fractional solution to theunderlying LP/QP relaxation and show its effectiveness in theory and practice.
Revised: Fractional Peeling : We introduce the idea of fractional peeling to round a fractional solution to theunderlying LP/QP relaxation and show its effectiveness in theory and practice.

Original: In experiments itsigniﬁcantly outperforms ordering based rounding algorithms considered previously.
Revised: In experiments itsigniﬁcantly outperforms ordering based rounding algorithms considered previously.

Original: We use it toobtain a provably approximate dense subgraph decomposition from a fractional solution.
Revised: We use it toobtain a provably approximate dense subgraph decomposition from a fractional solution.

Original: Connections to DSS: We explicitly connect the result of Fujishige [34] on the existence of a lexicographically optimal base in a polymatroid with the problem of computing a densest decomposition ofa supermodular function.
Revised: Connections to DSS: We explicitly connect the result of Fujishige [34] on the existence of a lexicographically optimal base in a polymatroid with the problem of computing a densest decomposition ofa supermodular function.

Original: We show that the a vertex load vector is feasible for the dual of Charikar’s
Revised: We show that the a vertex load vector is feasible for the dual of Charikar’s

Original: LP for DSG iff it is feasible for the base contrapolymatroid associated with the supermodular function f ( S ) =
Revised: LP for DSG iff it is feasible for the base contrapolymatroid associated with the supermodular function f ( S ) =

Original: | E ( S ) | induced by the graph.
Revised: | E ( S ) | induced by the graph.

Original: These connections clarify past results such as the one in[36, 2] and allow us to compare Frank-Wolfe versus projection based methods.
Revised: These connections clarify past results such as the one in[36, 2] and allow us to compare Frank-Wolfe versus projection based methods.

Original: Experimental evaluation: We compare ﬁve algorithms for DSG, namely G REEDY ++, F RANKW OLFE , MWU, FISTA, and FISTA-P ARALLEL a parallel version of FISTA.
Revised: Experimental evaluation: We compare ﬁve algorithms for DSG, namely G REEDY ++, F RANKW OLFE , MWU, FISTA, and FISTA-P ARALLEL a parallel version of FISTA.

Original: We test the algorithms on real world and synthetic data sets and show the effectiveness of FISTA when combinedwith fractional peeling.
Revised: We test the algorithms on real world and synthetic data sets and show the effectiveness of FISTA when combinedwith fractional peeling.

Original: To make the paper self contained we provide all proofs in the appendix, even for results implicitly contained in past literature.
Revised: To make the paper self contained we provide all proofs in the appendix, even for results implicitly contained in past literature.

Original: We focus on DSG (unweighted), however, the ideas generalize tohypergraphs and weighted graphs, and we leave a more in-depth exploration for future work.
Revised: We focus on DSG (unweighted), however, the ideas generalize tohypergraphs and weighted graphs, and we leave a more in-depth exploration for future work.

Original: 3. Densest Subgraph Decomposition and Supermodularity
Revised: 3. Densest Subgraph Decomposition and Supermodularity

Original: Let G =
Revised: Let G =

Original: ( V, E ) be an undirected graph.
Revised: ( V, E ) be an undirected graph.

Original: Each graph admits a unique nested decomposition of decreasing densities.
Revised: Each graph admits a unique nested decomposition of decreasing densities.

Original: This property is more transparently seen via supermodularity.
Revised: This property is more transparently seen via supermodularity.

Original: Let f : 2 V !
Revised: Let f : 2 V !

Original: R +be a non-negative supermodular set function 1 .
Revised: R +be a non-negative supermodular set function 1 .

Original: For example, ﬁx f ( S ) =
Revised: For example, ﬁx f ( S ) =

Original: | E ( S ) | which is supermodular.
Revised: | E ( S ) | which is supermodular.

Original: Supermodularity implies that there is a unique inclusion-wise maximal densest set S 1 .
Revised: Supermodularity implies that there is a unique inclusion-wise maximal densest set S 1 .

Original: It is convenient to describe the decomposition in an algorithmic fashion.
Revised: It is convenient to describe the decomposition in an algorithmic fashion.

Original: The algorithm calculates the maximal set S 1 ✓ V 0 =
Revised: The algorithm calculates the max- [Equation] imal set S 1 ✓ V 0 =

Original: V that maximizes | E ( S ) | | S | in G with density  1 =  ⇤ =
Revised: V that maximizes | E ( S ) | | S | in G with density  1 =  ⇤ =

Original: | E ( S 1 ) | | S 1 | .
Revised: | E ( S 1 ) | | S 1 | .

Original: For DSS this  f ( B ) for A ⇢ B corresponds to ﬁnding the unique maximal set S 1 that achieves the maximum density max S f ( S ) | S | .
Revised: For DSS thiscorresponds to ﬁnding the unique maximal set S 1 that achieves the maximum density max S f ( S ) | S | .

Original: In iteration i , letting U i  1 = [ 1  j<i S j , it calculates the maximal set S i ✓ V i
Revised: In iteration i , letting U i  1 = [ 1  j<i S j , it calculates the maximal set S i ✓ V i

Original: =
Revised: =

Original: V 
Revised: V  U

Original:  U i  1 that 140 maximizes ( | E ( S [ U i  1 |  | E ( U i  1 ) | ) / | S | .
Revised: i  1 that 141 maximizes ( | E ( S [ U i  1 |  | E ( U i  1 ) | ) / | S | .

Original: The algorithm is described formally in Algorithm 1.
Revised: The algorithm is described formally in Algorithm 1.

Original: U; , V
Revised: U; , V

Original: V , k while V k 6 = ; do k k +
Revised: V , k while V k 6 = ; do k k +

Original: S k arg max S ✓ V k  1 S maximal | E ( S )
Revised: S k arg max S ✓ V k  1 S maximal | E ( S )

Original: | + | E ( S,U k  1 )
Revised: | + | E ( S,U k  1 )

Original: | | S |
Revised: | | S |

Original: U k
Revised: U k

Original: U k  1 [ S k , V k
Revised: U k  1 [ S k , V k

Original: V k  1  S k return S 1 , ..., S k
Revised: V k  1  S k return S 1 , ..., S k

Original: U; , V
Revised: U; , V

Original: V , k while V k 6 = ; do k k +
Revised: V , k while V k 6 = ; do k k +

Original: S k arg max S ✓ V k  1 S maximal f ( S [ U k  1 )  f
Revised: S k arg max S ✓ V k  1 S maximal f ( S [ U k  1 )  f

Original: ( U k  1 ) | S |
Revised: ( U k  1 ) | S |

Original: U k
Revised: U k

Original: U k  1 [ S k , V k
Revised: U k  1 [ S k , V k

Original: V k  1  S k return S 1 , ..., S k
Revised: V k  1  S k return S 1 , ..., S k

Original: The algorithm outputs a partition of V into S 1 , ..., S k , where k is the dense decomposition depthof G (or any supermodular function f ).
Revised: The algorithm outputs a partition of V into S 1 , ..., S k , where k is the dense decomposition depthof G (or any supermodular function f ).

Original: With each set S i , we associate a density deﬁned as  i =| E ( S i ) | + | E ( S i ,U i  1 ) | | S i | (or in the case of f,  i =
Revised: With each set S i , we associate a density deﬁned as  i =| E ( S i ) | + | E ( S i ,U i  1 ) | | S i | (or in the case of f,  i =

Original: f ( S i
Revised: f ( S i

Original: [ U i  1 )  f
Revised: [ U i  1 )  f

Original: ( U i  1 ) | S i | ).
Revised: ( U i  1 ) | S i | ).

Original: In addition, for any u 2 S i , wesay the density of u,  u =  i .
Revised: In addition, for any u 2 S i , wesay the density of u,  u =  i .

Original: Refer to Lemma 6.2 (appendix) for some basic properties of the densedecomposition.
Revised: Refer to Lemma 6.2 (Appendix 6.1) for some basic properties of thedense decomposition.

Original: Speciﬁcally, the densities monotonically decrease, that is,  1 >  2 > ... >  k .
Revised: Speciﬁcally, the densities monotonically decrease, that is,  1 >  2 > ... > k .

Original: 4. LP and QP for DSG and Decomposition
Revised: 4. LP and QP for DSG and Decomposition

Original: Charikar’s [5] exact LP relaxation for DSG 4.1, and its dual 4.2 (in a slightly modiﬁed form), aregiven below.
Revised: Charikar’s [5] exact LP relaxation for DSG 4.1, and its dual 4.2 (in a slightly modiﬁed form), aregiven below.

Original: The primal has a variable y u for each vertex u 2 V which indicates whether u ischosen in the densest set.
Revised: The primal has a variable y u for each vertex u 2 V which indicates whether u ischosen in the densest set.

Original: For each edge e = { u, v } 2 E there is a variable z e to indicate whetherit is chosen.
Revised: For each edge e = { u, v } 2 E there is a variable z e to indicate whetherit is chosen.

Original: An edge { u, v } can be chosen only if both u and v are chosen which explains theconstraints linking z e to y u , y v .
Revised: An edge { u, v } can be chosen only if both u and v are chosen which explains theconstraints linking z e to y u , y v .

Original: The LP normalizes
Revised: The LP normalizes

Original: P u y u to 1 for linearity, and maximizes chosenedges.
Revised: P u y u to 1 for linearity, and maximizes chosenedges.max

Original: [Equation]
Revised: [Equation]

Original: The dual can be viewed as orienting each edge { u, v } fractionally towards u and v .
Revised: The dual can be viewed as orienting each edge { u, v } fractionally towards u and v .

Original: The orientationinduces loads on the vertices and the goal is to ﬁnd an orientation that minimizes the maximum loadover vertices.
Revised: The orientationinduces loads on the vertices and the goal is to ﬁnd an orientation that minimizes the maximum loadover vertices.

Original: It is not hard to see that LP 4.2 is actually the same as the dual of 4.1; the variablesb u can be replaced by a single variable b .
Revised: It is not hard to see that LP 4.2 is actually the same as the dual of 4.1; the variablesb u can be replaced by a single variable b .

Original: The optimum value of LP 4.1 and LP 4.2 is the density
Revised: The optimum value of LP 4.1 and LP 4.2 is the density

Original: ⇤ =
Revised: ⇤ =

Original:  1 .
Revised:  1 .

Original: However, we show that the b u values have additional information.
Revised: However, we show that the b u values have additional information.

Original: Theorem 4.1.
Revised: Theorem 4.1.

Original: Let S 1 , ..., S k and  1 , ...,  k be the densest subgraph decomposition of a graph G ,and for any u 2 S i , let  u
Revised: Let S 1 , ..., S k and  1 , ...,  k be the densest subgraph decomposition of a graph G ,and for any u 2 S i , let  u

Original: =  i .
Revised: =  i .

Original: There is an optimal solution ( x ⇤ , b ⇤ ) to 4.2 such that b ⇤ u
Revised: There is an optimal solution ( x ⇤ , b ⇤ ) to 4.2 such that b ⇤ u

Original: =  u .
Revised: =  u .

Original: We give two proofs in the appendix, one a direct proof using Rado’s theorem [37], and a secondby relating 4.2 to another relaxation and using a known result of Fujishige that is captured in Theorem 4.3.
Revised: We give two proofs, one a direct proof using Rado’s theorem [37] in Appendix 6.1, and a secondby relating 4.2 to another relaxation and using a known result of Fujishige that is captured in Theorem 4.3.

Original: We say a vector a 2 R n is lexicographically smaller than vector b 2 R n if the sortedvector a (in descending order) is lexicographically smaller than the sorted vector b .
Revised: We say a vector a 2 R n is lexicographically smaller than vector b 2 R n if the sortedvector a (in descending order) is lexicographically smaller than the sorted vector b .

Original: Theorem 4.1suggests that there exists a lexicographically least optimal solution to LP 4.2 where each vertex loadb u is precisely  u .
Revised: Theorem 4.1suggests that there exists a lexicographically least optimal solution to LP 4.2 where each vertex loadb u is precisely  u .

Original: To obtain the lexicographic solution, it sufﬁces to introduce some strict convexityinto the objective.
Revised: To obtain the lexicographic solution, it sufﬁces to introduce some strict convexityinto the objective.

Original: Let P ( x, b ) denote the polyhedron deﬁned by the constraints in 4.2.
Revised: Let P ( x, b ) denote the polyhedron deﬁned by the constraints in 4.2.

Original: Consider thequadratic program 4.3:
Revised: Consider the quadratic program 4.3:min

Original: [Equation]
Revised: [Equation]

Revised: such that ( x

Revised: [Equation]

Revised: b

Revised: [Equation]

Revised: [Equation]

Original: The theorem below was shown by Danisch et al.
Revised: The theorem below was shown by Danisch et al.

Original: [3] but as we will show later, this is a special caseof Fujishige’s result [34] from 1980 via Theorem 4.3.See Appendix.
Revised: [3] but as we will show later, this is a special caseof Fujishige’s result [34] from 1980 via Theorem 4.3.

Original: Theorem 4.2.
Revised: Theorem 4.2.

Original: There is a unique optimum solution b ⇤ to 4.3 and for each u 2 V ( G ) , b ⇤ u
Revised: There is a unique optimum solution b ⇤ to 4.3 and for each u 2 V ( G ) , b ⇤ u

Original: =  u .4.1.
Revised: =  u .4.1.

Original: LP and QP for DSS and densest decomposition
Revised: LP and QP for DSS and densest decomposition

Original: We will now generalize Theorem 4.1 and Theorem 4.2 for the DSS problem.
Revised: We will now generalize Theorem 4.1 and Theorem 4.2 for the DSS problem.

Original: We start by recappingthe notion of a contrapolymatroid (see [38]) which is the relevant notion for supermodular functions (as polymatroids are for submodular functions).
Revised: We start by recappingthe notion of a contrapolymatroid (see [38]) which is the relevant notion for supermodular functions (as polymatroids are for submodular functions).

Original: For a normalized non-negative supermodularfunction f : 2 V !
Revised: For a normalized non-negative supermodularfunction f : 2 V !

Original: R + , the contrapolymatroid with it is the following polyhedron
Revised: R + , the contrapolymatroid with it is the following polyhedron

Original: [Equation]
Revised: [Equation]

Original: A vector x 2 P f is a base if x ( V ) =
Revised: A vector x 2 P f is a base if x ( V ) =

Original: f ( V ) .
Revised: f ( V ) .

Original: The base contrapolymatroid is deﬁned as:
Revised: The base contrapolymatroid is deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: Now consider problem 4.6 where we are given a monotone non-negative supermodular function f : 2 V !
Revised: Now consider problem 4.6 where we are given a monotone non-negative supermodular function f : 2 V !

Original: R + and want to ﬁnd the lexicographically minimal solution b
Revised: R + and want to ﬁnd the lexicographically minimal solution b

Original: ⇤ for Problem 4.6.
Revised: ⇤ for Problem 4.6.

Original: We will show that we can do this by instead solving Problem 4.7:[Equation] minimize
Revised: We will show that we can do this by instead solving Problem 4.7: minimize

Original: [Equation]
Revised: [Equation]

Original: Theorem 4.3.
Revised: Theorem 4.3.

Original: Let S 1 , ..., S k and  1 , ...,  k be the densest supermodular set decomposition of f ,and for any u 2 S i , let  u
Revised: Let S 1 , ..., S k and  1 , ...,  k be the densest supermodular set decomposition of f ,and for any u 2 S i , let  u

Original: =  i .
Revised: =  i .

Original: Then the following must hold
Revised: Then the following must hold

Original: The solution b where b u =  u is feasible in the base polytope (i.e b 2 B f )
Revised: The solution b where b u =  u is feasible in the base polytope (i.e b 2 B f )

Original: The lexicographically minimal solution b ⇤ for Problem 4.6 satisﬁes b u =  u
Revised: The lexicographically minimal solution b ⇤ for Problem 4.6 satisﬁes b u =  u3.

Original: The optimal solution of 4.7, b ⇤ is unique, and for each u 2 V, b ⇤ u
Revised: The optimal solution of 4.7, b ⇤ is unique, and for each u 2 V, b ⇤ u

Original: =  u
Revised: =  u

Original: We give a self-contained proof in the appendix and note that the theorem is implied by (essentiallyequivalent to)
Revised: We give a self-contained proof in Appendix 6.1 and note that the theorem is implied by (essentially equivalent to)

Original: Fujishige’s result [34] on the existence of a lexicographically optimal base of a polymatroid with respect to a weight vector.
Revised: Fujishige’s result [34] on the existence of a lexicographically optimal base ofa polymatroid with respect to a weight vector.

Original: LP 4.2 and LP 4.6 can be related via the following theorem.
Revised: LP 4.2 and LP 4.6 can be related via the followingtheorem.

Original: Theorem 4.4.
Revised: Theorem 4.4.

Original: Consider a graph G =
Revised: Consider a graph G =

Original: ( V, E ) and the associated supermodular function f : 2 V !
Revised: ( V, E ) and the associated supermodular function f : 2 V !

Original: R + where f ( S ) =
Revised: R + where f ( S ) =

Original: | E ( S ) | .
Revised: | E ( S ) | .

Original: A vector b 2 B f if and only if there is an x 2 R ord ( E ) , x  0 such thatthe pair ( x, b ) satisfy the constraints of the LP 4.2.
Revised: A vector b 2 B f if and only if there is an x 2 R ord ( E ) , x  0 such thatthe pair ( x, b ) satisfy the constraints of the LP 4.2.

Original: See appendix for a proof.
Revised: See Appendix 6.1 for a proof.

Original: This implies that Theorem 4.2 is a corollary of Theorem 4.3.
Revised: This implies that Theorem 4.2 is a corollary of Theorem 4.3.

Original: 5. Solving the Quadratic Program using proximal projections, and rounding
Revised: 5. Solving the Quadratic Program using proximal projections, and rounding

Original: In this section we show how to approximately solve 4.3.
Revised: In this section we show how to approximately solve 4.3.

Original: We let f ( x ) =
Revised: We let f ( x ) =

Original: P u 2 V ⇣ P v 2  ( u ) x uv ⌘ 2 .
Revised: P u 2 V ⇣P v 2  ( u ) x uv ⌘ 2 .

Original: Note that this is simply the objective function rewritten in terms of x .
Revised: Note that this is simply the objective function rewritten in terms of x .

Original: Similarly, let h ( x ) be anindicator function where h ( x ) =
Revised: Similarly, let h ( x ) be anindicator function where h ( x ) =

Original: if x uv  0 and x uv + x vu = 1 , 8 ( u, v ) 2 E and + 1 otherwise.
Revised: if x uv  0 and x uv + x vu = 1 , 8 ( u, v ) 2 E and + 1 otherwise.

Original: Then Problem 4.3 can be rewritten as minimizing the unconstrained objective f ( x ) + h ( x ) forx 2 R 2 m .
Revised: Then Problem 4.3 can be rewritten as minimizing the unconstrained objective f ( x ) + h ( x ) forx 2 R 2 m .

Original: We will use a proximal gradient method to solve the problem.
Revised: We will use a proximal gradient method to solve the problem.

Original: For that, we need twolemmas.
Revised: For that, we need twolemmas whose proofs are in Appendix 6.2.

Original: Lemma 5.1.
Revised: Lemma 5.1.

Original: The Lipschitz constant of r f is at most 2  ( G ) where  ( G ) is the max degree of G .
Revised: The Lipschitz constant of r f is at most 2  ( G ) where  ( G ) is the max degree of G .

Original: Lemma 5.2.
Revised: Lemma 5.2.

Original: Let x 2 R 2 m .
Revised: Let x 2 R 2 m .

Original: Deﬁne the proximal mapping prox h
Revised: Deﬁne the proximal mapping prox h

Original: ( x ) as the point p 2 R 2 m thatminimizes k p  x k 2 such that h ( p ) = 0 .
Revised: ( x ) as the point p 2 R 2 m thatminimizes k p  x k 2 such that h ( p ) = 0 .

Original: Then we have that for u < v , prox h
Revised: Then we have that for u < v , prox h

Original: ( x ) uv = x uv  x vu +1 2if | x uv  x vu |  1 , prox h
Revised: ( x ) uv = x uv  x vu +1 2if | x uv  x vu |  1 , prox h

Original: ( x ) uv
Revised: ( x ) uv

Original: = 1 if x uv  x vu > 1 , and prox h
Revised: = 1 if x uv  x vu > 1 , and prox h

Original: ( x ) uv = 

Revised: ( x ) uv =

Original: if x uv  x vu <  1 .[-
-]
Revised: if x uv  x vu <  1 .[-
-]

Original: Additionally, prox h ( x ) vu = 1  prox h
Revised: Additionally, prox h ( x ) vu = 1  prox h

Original: ( x ) uv .
Revised: ( x ) uv .

Original: We present the algorithm now.
Revised: We present the algorithm now.

Original: We are interested in the unconstrained optimization problem ofminimizing f ( x ) + h ( x ) where f is convex, and h has an easy to compute proximal mapping.
Revised: We are interested in the unconstrained optimization problem ofminimizing f ( x ) + h ( x ) where f is convex, and h has an easy to compute proximal mapping.

Original: This type of problem can be solved using proximal gradient methods.
Revised: Thistype of problem can be solved using proximal gradient methods.

Original: From a high level, the (basic)algorithm is described Algorithm 2.
Revised: From a high level, the (basic)algorithm is described Algorithm 2.

Original: At any iteration t it has a guess for the minimizer x
Revised: At any iteration t it has a guess for the minimizer x

Original: ( t ) .
Revised: ( t ) .

Original: It thencalculates the gradient of f and moves slightly against it.
Revised: It thencalculates the gradient of f and moves slightly against it.

Original: However, since this might make the newguess infeasible, it uses the proximal mapping to project the new guess to a feasible solution.
Revised: However, since this might make the newguess infeasible, it uses the proximal mapping to project the new guess to a feasible solution.

Original: Algorithm
Revised: Input: f and h with prox h ( x ) , learning rate ↵ and iterations T . Initialize x (0) with h ( x (0) ) =for t 2 [1 , T ] do x (

Original: [Equation]
Revised: [Equation]

Original: for do
Revised: ) = prox

Original: [Equation]
Revised: [Equation]

Revised: r f ( x ( t  1) )) return x

Revised: ( T ) y (0)

Revised: =

Revised: x (0) for t 2 [1 , T ] do x ( t ) =

Original: While the basic proximal gradient method works, we will use an even faster (both theoreticallyand practically) version known as the accelerated proximal gradient method which incorporates
Revised: prox h ( y ( t  1)  ↵ r f ( y ( k  1) )) y ( t ) = x ( t ) + t  1 t +2 ( x ( t )  x ( t  1) ) return x

Revised: ( T )

Revised: While the basic proximal gradient method works, we will use an even faster (both theoreticallyand practically) version known as the accelerated proximal gradient method which incorporates

Original: Nesterov-like momentum terms [39] in the projection step.
Revised: Nesterov-like momentum terms [39] in the projection step.

Original: It has other names in the literaturesuch as proximal gradient method with extrapolation and FISTA [40].
Revised: It has other names in the literaturesuch as proximal gradient method with extrapolation and FISTA [40].

Original: The algorithm is outlined in
Revised: The algorithm is outlined in

Original: Algorithm 2.
Revised: Algorithm 2.

Original: We have the following known result on the FISTA algorithm.
Revised: We have the following known result on the FISTA algorithm.

Original: Lemma 5.3.
Revised: Lemma 5.3.

Original: Let x ⇤ be the minimizer of f .
Revised: Let x ⇤ be the minimizer of f .

Original: Suppose that the learning rate satisﬁes ↵  1 L ( f )where L ( f ) is the Lipschitz constant of r f .
Revised: Suppose that the learning rate satisﬁes ↵  1 L ( f )where L ( f ) is the Lipschitz constant of r f .

Original: Then after k iterations, f ( x ( k ) )
Revised: Then after k iterations, f ( x ( k ) )

Original:  f( x ⇤ )  2 k x (0)  x ⇤ k 2 ↵ k. In our case, we bounded the Lipschitz constant to 2  ( G ) and an easy upper bound on  x (
Revised:  f

Original: [Equation]
Revised: [Equation]

Original:  )   x ⇤  2is 2 m = 2 | E ( G ) | .
Revised: In our case, we bounded the Lipschitz constant to 2  ( G ) and an easy upper bound on  x (0)  x ⇤  2is 2 m = 2 | E ( G ) | .

Original: Combining this with a learning rate of 1 2  ( G ) , we get the following result
Revised: Combining this with a learning rate of 1 2  ( G ) , we get the following result

Original: Lemma 5.4.
Revised: Lemma 5.4.

Original: If FISTA is applied to the objective f ( x ) + h ( x ) as previously deﬁned, then in the k thiteration, we have f ( x ( k ) )
Revised: If FISTA is applied to the objective f ( x ) + h ( x ) as previously deﬁned, then in the k thiteration, we have f ( x ( k ) )

Original:  f ( x ⇤ )
Revised:  f ( x ⇤ )

Original:  8  ( G ) m k.
Revised:  8  ( G ) m k.

Original: The ﬁnal FISTA algorithm for our problem is shown in Algorithm 3 inthe Appendix  with the correctgradient computation, projective mappings, and iteration updates.
Revised: The ﬁnal FISTA algorithm for our problem is shown in Algorithm 3 in Appendix 6.2 with the correctgradient computation, projective mappings, and iteration updates.

Original: Lastly, to get a good additiveapproximation on each  u , we have the following result {+whose proof is in Appendix 6.
+}
Revised: Lastly, to get a good additiveapproximation on each  u , we have the following result {+whose proof is in Appendix 6.
+}

Original: Theorem 5.5.
Revised: Theorem 5.5.

Original: Let b ⇤ u
Revised: Let b ⇤ u

Original: =  u .
Revised: =  u .

Original: After t
Revised: After t

Original: =
Revised: =

Original: O ( p  ( G ) m ✏ ) iterations of FISTA, we must have
Revised: O ( p  ( G ) m ✏ ) iterations of FISTA, we must have

Original: b ( t )  b ⇤   ✏ .
Revised: b ( t )  b ⇤   ✏ .

Original: Thistrivially implies 
Revised: This implies  b

Original: [Equation]
Revised: [Equation]

Revised: t

Original:  for all u 2 V
Revised: [Equation] u   [Equation]   ✏ for all u 2 V5.1.

Original: 5.1. Fractional Peeling.
Revised: Fractional Peeling.

Original: Given an approximate solution ( b, x ) to Problem 4.3, how do we round it to obtain a good densestdecomposition?
Revised: Given an approximate solution ( b, x ) to Problem 4.3, how do we round it to obtain a good densestdecomposition?

Original: The natural approach is to sort the vertices in non-increasing values of b and takesufﬁxes.
Revised: The natural approach is to sort the vertices in non-increasing values of b and takesufﬁxes.

Original: This is used in [36].
Revised: This is used in [36].

Original: This is an exact rounding algorithm when b is an optimum solution,however one can construct approximate solutions for which this rounding is not ideal.
Revised: This is an exact rounding algorithm when b is an optimum solution,however one can construct approximate solutions for which this rounding is not ideal.

Original: We describea peeling algorithm inspired by G REEDY ++ [1] that takes advantage of the auxiliary informationprovided by the vector x .
Revised: We describea peeling algorithm inspired by G REEDY ++ [1] that takes advantage of the auxiliary informationprovided by the vector x .

Original: First, set b 

Revised: First, set b

Original: =
Revised: =

Original: b and G (0) =
Revised: b and G (0) =

Original: G .
Revised: G .

Original: In iteration, t  1 , peel the vertex u withminimum current load b 0 u .
Revised: In iteration, t  1 , peel the vertex u withminimum current load b 0 u .

Original: Then, for each v 2  G t  1 ( u ) , set b 0 v b 0 v  x vu (i.e subtract the fractionalvalue of x vu from v ’s load).
Revised: Then, for each v 2  G t  1 ( u ) , set b 0 v b 0 v  x vu (i.e subtract the fractionalvalue of x vu from v ’s load).

Original: Update G t =
Revised: Update G t =

Original: G t  1  u .
Revised: G t  1  u .

Original: Repeat this process to obtain G (0) , ..., G ( n ) .
Revised: Repeat this process to obtain G (0) , ..., G ( n ) .

Original: Finally, return a graph G ( i ) with maximum density.
Revised: Finally, return a graph G ( i ) with maximum density.

Original: The algorithm can be implemented in O ( m +n log n ) using a Fibonacci heap.
Revised: The algorithm can be implemented in O ( m +n log n ) using a Fibonacci heap.

Original: We will refer to this process as fractional peeling .
Revised: We will refer to this process as fractional peeling .

Original: We show boththeoretically and experimentally that fractional peeling leads to better algorithms.
Revised: We show boththeoretically and experimentally that fractional peeling leads to better algorithms.

Original: 5.2. ✏  dense local decomposition
Revised: 5.2. ✏  dense local decomposition

Original: We show that fractional peeling can be used on an approximate solution to 4.3 to obtain an approximate dense decomposition with a theoretical guarantee.
Revised: We show that fractional peeling can be used on an approximate solution to 4.3 to obtain an approximate dense decomposition with a theoretical guarantee.

Original: This is in contrast to previous work in [36].
Revised: This is in contrast to previous work in [36].

Original: We deﬁne a strong notion of approximate decomposition.
Revised: We deﬁne a strong notion of approximate decomposition.

Original: Given a solution ( b, x ) to 4.3, we say ˆ b is an ✏ -load-vector if  ˆ b  b ⇤   ✏ .
Revised: Given a solution ( b, x ) to 4.3, we say ˆ b is an ✏ -load-vector if  [Equation]   ✏ .

Original: Given a partition of the vertices T 1 , ..., T r , we say the partition is an ✏ -approximate dense decomposition to S 1 , ..., S k
Revised: Given a partition of the vertices T 1 , ..., T r , we say the partition is an ✏ -approximate dense decomposition to S 1 , ..., S k

Original: (the true dense decomposition)
Revised: (the true dense decomposition)

Original: if
Revised: if

Original: [Equation]
Revised: [Equation]

Original:  ✏

Original: Intuitively, what this says is that every u 2 V belongs to a set T h that has a density that is not muchless than  u .
Revised: Intuitively, what this says is that every u 2 V belongs to a set T h that has a density that is not muchless than  u .

Original: Theorem 5.6.
Revised: Theorem 5.6.

Original: Given an ✏ load vector b and an edge vector x that induces b , we can calculate an✏  p n  -approximate dense decomposition in ˜ O
Revised: Given an ✏ load vector b and an edge vector x that induces b , we can calculate an✏ ( p n + 1) -approximate dense decomposition in ˜ O

Original: ( mn ) time.
Revised: ( mn ) time.

Revised: The proof is in Appendix 6.4.

Original: Note that the notion of error is additive and holds for every vertex u .
Revised: Note that the notion of error is additive and holds for every vertexu .

Original: Although the p n -factor islarge, the analysis shows that one can usually obtain a much stronger bound.
Revised: Although the p n -factor is large, the analysis shows that one can usually obtain a much strongerbound.

Original: Qualitatively it showsthat fractional peeling leads to good dense decomposition as " ! 0 .
Revised: Qualitatively it shows that fractional peeling leads to good dense decomposition as " ! 0 .

Original: 5.3. Projections vs Frank-Wolfe vs MWU for DSG and DSS
Revised: 5.3. Projections vs Frank-Wolfe vs MWU for DSG and DSS

Original: The Frank-Wolfe method is natural to apply to solve 4.7 since each iteration requires optimizing alinear objective over the base contrapolymatroid B f ; this is easy and fast via the greedy algorithm,as shown originally by Edmonds in the context of polymatroids (see [38]).
Revised: The Frank-Wolfe method is natural to apply to solve 4.7 since each iteration requires optimizing alinear objective over the base contrapolymatroid B f ; this is easy and fast via the greedy algorithm,as shown originally by Edmonds in the context of polymatroids (see [38]).

Original: Danisch et al.
Revised: Danisch et al.

Original: workwith 4.3.
Revised: workwith 4.3.

Original: However, as we observed earlier, 4.3 is a compact way to represent the associated basepolyhedron, and hence the algorithm in [36] is the same as the Frank-Wolfe algorithm applied to4.7.
Revised: However, as we observed earlier, 4.3 is a compact way to represent the associated basepolyhedron, and hence the algorithm in [36] is the same as the Frank-Wolfe algorithm applied to4.7.

Original: The Fujishige-Wolfe minimum norm point algorithm for submodular function minimization isrelated but is based on Wolfe’s method (see [41, 42]).
Revised: The Fujishige-Wolfe minimum norm point algorithm for submodular function minimizationis related but is based on Wolfe’s method (see [41, 42]).

Original: The fact that the optimum point in the basepolytope has additional information was already pointed out in [34], and also explored in the context of size-constrained submodular function minimization by Nagano et al.
Revised: The fact that the optimum point in thebase polytope has additional information was already pointed out in [34], and also explored in thecontext of size-constrained submodular function minimization by Nagano et al.

Original: The advantage ofproximal gradient methods such as FISTA is their faster convergence rates when compared to the
Revised: The advantageof proximal gradient methods such as FISTA is their faster convergence rates when compared to the

Original: Frank-Wolfe method, although each iteration requires a projection oracle for B f .
Revised: Frank-Wolfe method, although each iteration requires a projection oracle for B f .

Original: Our algorithm isbased on the observation that there is an O ( m ) -time projection oracle for DSG due to the alternatecharacterization of B f via the edge variable LP 4.2.
Revised: Our algorithm isbased on the observation that there is an O ( m ) -time projection oracle for DSG due to the alternatecharacterization of B f via the edge variable LP 4.2.

Original: This same observation holds for hypergraphsand we obtain a fast algorithm for them.
Revised: This same observation holds for hypergraphsand we obtain a fast algorithm for them.

Original: We outline the formal details in the appendix.
Revised: We outline the formal details in Appendix 6.5 and 6.6.

Original: The multiplicative weight update (MWU) method is another broad methodology for solving linear and convexprograms and there are several variants.
Revised: The multiplicative weight update (MWU) method is another broad methodology for solving linearand convex programs and there are several variants.

Original: One variant of MWU for solving 4.2 can be interpreted asa Frank-Wolfe method with a certain convex objective and with a certain step size; each iteration ofthis algorithm is again a greedy optimization over B f .
Revised: One variant of MWU for solving 4.2 can beinterpreted as a Frank-Wolfe method with a certain convex objective and with a certain step size;each iteration of this algorithm is again a greedy optimization over B f .

Original: We provide more details in the Appendix 6.7.5.4.
Revised: We provide more details inthe Appendix 6.8.

Original:  A theoretically fast algorithm for approximate load vector via min-cost ﬂow
Revised: 5.4. A theoretically fast algorithm for approximate load vector via min-cost ﬂow

Original: While Algorithm 3 (detailed in the appendix) works extremely well in practice, there is room to improve the theoretical run time.
Revised: While Algorithm 3 (detailed in Appendix 6.2 ) works extremely well in practice, there is room to improve the theoretical run time.

Original: Each iteration takes O ( m ) time and can easily be parallelized.
Revised: Each iteration takes O ( m ) time and can easily be parallelized.

Original: However, in the worst-case it would need O ( m p  ( G ) m/ ✏ ) time to get an additive ✏ -approximation.
Revised: However, in the worst-case it would need O ( m p  ( G ) m/ ✏ ) time to get an additive ✏ -approximation.

Original: Can we obtain a theoretically faster algorithm?
Revised: Can we obtain a theoretically faster algorithm?

Original: Inthe Appendix  we describe a reduction to computing exact minimum-cost ﬂow in a directed network.
Revised: In Appendix 6.7 we describe a reduction to computing exact minimum-cost ﬂow in a directed network.

Original: Via the new breakthrough near-linear time algorithm for min-cost ﬂows [43], we obtain an O ( m p n ✏ ) 

Revised: Via the new breakthrough near-linear time algorithm for min-cost ﬂows [43], we obtain an O ( m p n ✏ )

Original: + o (1) ) time algorithm to get an ✏ -approximateload vector.
Revised: + o (1) ) time algorithm to get an ✏ -approximateload vector.

Original: We hope that this theoretical result will inspire the development of new algorithms thatare provably faster in theory while also having good empirical performance.
Revised: We hope that this theoretical result will inspire the development of new algorithms thatare provably faster in theory while also having good empirical performance.

Original: 6. Experimental Evaluation
Revised: 6. Experimental Evaluation

Original: Datasets & Implementation Details.
Revised: Datasets & Implementation Details.

Original: We ran experiments on 7 real world datasets ( 6 from the
Revised: We ran experiments on 7 real world datasets ( 6 from the

Original: SNAP database [44], and 1 from [45]), and one tailored synthetic dataset (used for clarifying animportant difference between all algorithms) for a total of 8 datasets.
Revised: SNAP database [44], and 1 from [45]), and one tailored synthetic dataset (used for clarifying animportant difference between all algorithms) for a total of 8 datasets.

Original: The dataset information issummarized in the table below.
Revised: The dataset information issummarized in the table below.

Original: The C LOSE -C LIQUES dataset consists of the complete bipartitegraph K d,D for d = 30 and D = 2000 , and 20 copies of the complete graph K h where h = 60 .
Revised: The C LOSE -C LIQUES dataset consists of the complete bipartitegraph K d,D for d = 30 and D = 2000 , and 20 copies of the complete graph K h where h = 60 .

Original: We tested 5 algorithms to approximate DSG: The FISTA based algorithm (sequential, no parallelism), FISTA-P ARALLEL , G REEDY ++ from [1], the F RANK -W OLFE based algorithm from [3],and the MWU algorithm respectively. See supplementary section for implementation details of the 5algorithms. Also see appendix for details on the speciﬁc variant of MWU we tested. All algorithmswere implemented in C++17 and were compiled with O3 and UNROLL - LOOPS optimizations.
Revised: We tested 5 algorithms to approximate DSG: The FISTA based algorithm (sequential, no parallelism), FISTA-P ARALLEL , G REEDY ++ from [1], the F RANK -W OLFE based algorithm from [3],

Original: The

Original: Dataset
Revised: Dataset

Original: Vertices
Revised: Vertices

Original: Edges
Revised: Edges

Original: Source
Revised: Source

Original: Dataset
Revised: Dataset

Original: Vertices
Revised: Vertices

Original: Edges
Revised: Edges

Original: Source cit-Patents 3,774, 16,518, [44] roadNet-CA 1,965, 2,766, [44] com-Amazon 334, 925, [44]
Revised: Source cit-Patents 3,774, 16,518, [44] roadNet-CA 1,965, 2,766, [44] com-Amazon 334, 925, [44]

Original: Close-Cliques 3, 95,
Revised: Close-Cliques 3, 95,

Original: Synthetic orkut 3,072, 117,185, [45] dblp-author 317, 1,049, [44] roadNet-PA 1,088, 1,541, [44] wiki-topcats 1,791, 28,508, [44]
Revised: Synthetic orkut 3,072, 117,185, [45] dblp-author 317, 1,049, [44] roadNet-PA 1,088, 1,541, [44] wiki-topcats 1,791, 28,508, [44]

Original: [Figure]
Revised: [Figure]

Revised: and the MWU algorithm respectively.

Revised: See supplementary section for implementation details of the5 algorithms.

Revised: Also see Appendix 6.8 for details on the speciﬁc variant of MWU we tested.

Revised: All algorithms were implemented in C++17 and were compiled with O3 and UNROLL - LOOPS optimizations.

Original:  implementations of Algorithms FISTA, FISTA-P ARALLEL , F RANK -W OLFE , and MWU are theauthors’ implementations, but we used the original implementation for G REEDY ++ [1] as it wasextremely well optimized.
Revised: The implementations of Algorithms FISTA, FISTA-P ARALLEL , F RANK -W OLFE , and MWU arethe authors’ implementations, but we used the original implementation for G REEDY ++ [1] as it wasextremely well optimized.

Original: We modiﬁed their implementation minimally to log basic informationneeded for the evaluation.
Revised: We modiﬁed their implementation minimally to log basic informationneeded for the evaluation.

Original: FISTA-P ARALLEL used Open MPI [46] for parallelism.
Revised: FISTA-P ARALLEL used Open MPI [46] for parallelism.

Original: We ran ourexperiments on a Slurm-based university campus cluster.
Revised: We ran ourexperiments on a Slurm-based university campus cluster.

Original: For all machines, we requested 1 node and16 cores per experiment.
Revised: For all machines, we requested 1 node and16 cores per experiment.

Original: The nodes had 64 GB of RAM and Xeon PHI 5100 CPUs.
Revised: The nodes had 64 GB of RAM and Xeon PHI 5100 CPUs.

Original: Densities .
Revised: Densities .

Original: In the ﬁrst experiment, we ran all the algorithms for a number of iterations, and monitoredthe maximum density reached by the algorithm until iteration t .
Revised: In the ﬁrst experiment, we ran all the algorithms for a number of iterations, and monitoredthe maximum density reached by the algorithm until iteration t .

Original: The density in each iteration wascalculated by statically sorting the vertices by the value of their load vector, and peeling in that orderand returning the maximum density subgraph in that iteration.
Revised: The density in each iteration wascalculated by statically sorting the vertices by the value of their load vector, and peeling in that orderand returning the maximum density subgraph in that iteration.

Original: The result is shown in Figure 6.1(Top 4 plots, we only show 4 datasets, remaining plots in appendix).
Revised: The result is shown in Figure 6.1 (Top4 plots, we only show 4 datasets, remaining plots in Appendix 6.8).

Original: Speciﬁcally, this does not usethe fractional peeling technique we discussed.
Revised: Speciﬁcally, this does not usethe fractional peeling technique we discussed.

Original: We also plot the maximum density reached by usingfractional peeling instead (Bottom 4 plots in 6.1, we show 4 datasets, remaining plots in appendix).
Revised: We also plot the maximum density reached by usingfractional peeling instead (Bottom 4 plots in 6.1, we show 4 datasets, remaining plots in Appendix6.8).

Original: When the peeling order is changed from a static peel (based on the load vector value) to a fractional peel, the results improve dramatically.
Revised: When the peeling order is changed from a static peel (based on the load vector value) to a fractionalpeel, the results improve dramatically.

Original: See Figure 6.1 and appendix.
Revised: See Figure 6.1 and Appendix 6.8.

Original: In all algorithms (except
Revised: In all algorithms (except

Original: MWU), fractional peeling leads to a dramatic speedup in terms of the number of iterations needed toget the densest subgraph on almost all datasets.
Revised: MWU), fractional peeling leads to a dramatic speedup in terms of the number of iterations needed toget the densest subgraph on almost all datasets.

Original: Fractional peeling provides little beneﬁt to MWU.
Revised: Fractional peeling provides little beneﬁt to MWU.

Original: We can observe that for all the real world datasets, G REEDY ++ and FISTA are extremely competitive and reach near-optimal densities in just a few iterations.
Revised: We can observe that for all the real world datasets, G REEDY ++ and FISTA are extremely competitive and reach near-optimal densities in just a few iterations.

Original: Meanwhile, the F RANK -W OLFE basedalgorithm lagged behind in the beginning, but steadily made progress towards the maximum densityin later iterations.
Revised: Meanwhile, the F RANK -W OLFE basedalgorithm lagged behind in the beginning, but steadily made progress towards the maximum densityin later iterations.

Original: MWU strongly lagged behind all algorithms in several datasets.
Revised: MWU strongly lagged behind all algorithms in several datasets.

Original: The only exception is for the C LOSE -C LIQUES dataset (see Appendix).
Revised: The only exception is for the C LOSE -C LIQUES dataset (see Appendix 6.8).

Original: The component in the densest subgraph hasdensity (30 ⇥ 2000) / (30 + 2000) ⇡ 29 . 5566 .
Revised: The component in the densest subgraphhas density ⇡ 29 . 5566 .

Original: Meanwhile, the remaining components have density  / 60 = 29 . 
Revised: Meanwhile, the remaining components have density   / 60 = 29 . 5 .

Original: FISTA, F RANK -W OLFE , and MWU eventually found the densest subgraph inthe ﬁrst 100 iterations, but G REEDY ++ was stuck in a local optima for 200 iterations!
Revised: This is a similar but nonidentical to the example from [1].

Original: Note that theexample K d,D and r copies of K 2 d (for sufﬁciently large r, D ) can be used to theoretically showthat G REEDY ++ requires ⌦ ( 1 ✏ ) iterations to converge to a (1  ✏ ) approximation for DSG.
Revised: Note that for K d,D and r copies of K 2 d (forsufﬁciently large r, D ), 9 ✏ where G REEDY ++ requires ⇥ ( 1 ✏ ) iterations to converge to a (1  ✏ ) approximation for DSG in the worst case.

Original: MWU and F RANK -W
Revised: MWU and F RANK -W

Original: OLFE did better for this synthetic example where the densities are very close and itis worth understanding in more detail.
Revised: OLFE did better for this syntheticexample where the densities are very close and it is worth understanding in more detail.

Original: Wall clock time and run-time per iteration.
Revised: Wall clock time and run-time per iteration.

Original: In the second experiment we examine the wallclock time and time per iteration.
Revised: In the second experiment we examine the wall-clocktime and time per iteration.

Original: We study the maximum density reached by an algorithm after Tseconds of wall-clock time (cumulative time of all iterations).
Revised: We study the maximum density reached by an algorithm after T secondsof wall-clock time (cumulative time of all iterations).

Original: Figure 6.12 (appendix) and last 4 plotsof Figure 6.2 shows the result.
Revised: Figure 6.15 (Appendix 6.8) and last 4 plotsof Figure 6.2 shows the result.

Original: As can be seen, FISTA-P ARALLEL and FISTA (sequential) ﬁndsthe maximum density on almost all datasets in the least wall-clock time (albeit sequential FISTAand F RANK -W OLFE are close runners if we only restrict to non-parallel implementations).
Revised: As can be seen, FISTA-P ARALLEL and FISTA (sequential) ﬁndsthe maximum density on almost all datasets in the least wall-clock time (albeit sequential FISTAand F RANK -W OLFE are close runners if we only restrict to non-parallel implementations).

Original: On theother hand, MWU performed poorly in wall-clock time, often taking orders of magnitude longerthan FISTA or G REEDY ++ to ﬁnd suboptimal dense subgraphs.
Revised: On theother hand, MWU performed poorly in wall-clock time, often taking orders of magnitude longerthan FISTA or G REEDY ++ to ﬁnd suboptimal dense subgraphs.

Original: Overall, the number of iterationsof FISTA is the lowest, and each iteration is fast and can be parallelized, so it is the best performer.     
Revised: Overall, the number of iterationsof FISTA is the lowest, and each iteration is fast and can be parallelized, so it is the best performer.     10 Wallclocktime(seconds) 1 .1 .1 .1 .1 .1 . D e n s it y

Revised: DensityforRoadnetCA ﬁstaseqﬁstaparallelfrankwolfegreedyppmwu

Original: [Figure]
Revised: [Figure]

Original: (seconds) [-
-]
Revised: (seconds) [-
-]1 .1 .1 .1 .1 .

Original: D e n s it y
Revised: D e n s it y

Revised: [Figure]

Original: DensityforWikipediaTopCategories ﬁstaseqﬁstaparallelfrankwolfegreedyppmwu 
Revised: ﬁstaseqﬁstaparallelfrankwolfegreedyppmwu [Figure] [Figure] [Table]

Revised: [Figure]

Original: We also considered the run time per iteration for the algorithms.
Revised: We also considered the run time per iteration for the algorithms.

Original: Figure 6.8 (appendix) shows theresult.
Revised: Figure 6.11 (appendix) shows theresult.

Original: The per iteration time depends on several factors.
Revised: The per iteration time depends on several factors.

Original: Although all algorithms have linear or nearlinear in m running time, the speciﬁc data structure used and cache performance can have substantialimpact.
Revised: Although all algorithms have linear or nearlinear in m running time, the speciﬁc data structure used and cache performance can have substantialimpact.

Original: Overall, F RANK -W OLFE (due to its simple implementation) and FISTA-P ARALLEL hadthe best per-iteration performance.
Revised: Overall, F RANK -W OLFE (due to its simple implementation) and FISTA-P ARALLEL hadthe best per-iteration performance.

Original: The average speedup per iteration from FISTA-P ARALLEL over
Revised: The average speedup per iteration from FISTA-P ARALLEL over

Original: G REEDY ++ was roughly 5 fold.
Revised: G REEDY ++ was roughly 5 fold.

Original: Note that due to the peeling nature of G REEDY ++, it cannot beparallelized like FISTA.
Revised: Note that due to the peeling nature of G REEDY ++, it cannot beparallelized like FISTA.

Original: We believe that there is further room to improve the memory and cacheperformance of the FISTA-based algorithm by using only one variable for edge rather than two.
Revised: We believe that there is further room to improve the memory and cacheperformance of the FISTA-based algorithm by using only one variable for edge rather than two.

Original: This can further improve the overall wall-clock time of FISTA and FISTA-P ARALLEL .
Revised: This can further improve the overall wall-clock time of FISTA and FISTA-P ARALLEL .

Original: Convergence to optimal load vector.
Revised: Convergence to optimal load vector.

Original: Recall that we are minimizing
Revised: Recall that we are minimizing

Original: P u b 2 u , and hence the norm b ( t )  gives a proxy for the convergence of the errors in b ( t ) .
Revised: P u b 2 u , and hence the norm  b ( t )   gives a proxy for the convergence of the errors in b ( t ) .

Original: In this experiment, we plotted the normof the vector b ( t ) for each algorithm in each dataset.
Revised: In this experiment, we plotted thenorm of the vector b ( t ) for each algorithm in each dataset.

Original: The result is shown in Figure 6.9 (Appendix).
Revised: The result is shown in Figure 6.12 (Appendix 6.8).

Original: G REEDY ++ does very well in the ﬁrst few iterations, but slows down in its improvement on  b ( t )  as the number of iterations increases.
Revised: G REEDY ++ does very well in the ﬁrst few iterations, but slows down in its improvement on [Equation]   as the number of iterations increases.

Original: In comparison, FISTA eventually surpasses G REEDY ++ andreduces the error at a faster rate than G REEDY ++.
Revised: In comparison, FISTA eventually surpasses G REEDY ++ and reduces the error at a faster rate than G REEDY ++.

Original: In comparison, F RANK -W OLFE and MWU startwith a substantial error in the b vector but quickly reduce it, however, their error was always worsethat both G REEDY ++ and FISTA in all iterations even when left to runup to 200 iterations.
Revised: In comparison, F RANK -W OLFEand MWU start with a substantial error in the b vector but quickly reduce it, however, their error wasalways worse that both G REEDY ++ and FISTA in all iterations even when left to run 200 iterations.

Original: Conclusion.
Revised: Conclusion.

Original: We introduced a new iterative algorithm for the densest subgraph and densest decomposition problems.
Revised: We introduced a new iterative algorithm for the densest subgraph and densest decomposition problems.

Original: We also described a new fractional peeling technique which has strong empiricalperformance as well as theoretical guarantees and showed experimentally how it improved almostall existing algorithms compared to static load sorting.
Revised: We also described a new fractional peeling technique which has strong empiricalperformance as well as theoretical guarantees and showed experimentally how it improved almostall existing algorithms compared to static load sorting.

Original: The new algorithm is scalable and simple,and can be applied to graphs with hundreds of millions of edges.
Revised: The new algorithm is scalable and simple,and can be applied to graphs with hundreds of millions of edges.

Original: Our experiments support thetheory established on the utility of the new algorithm and fractional peeling.
Revised: Our experiments support thetheory established on the utility of the new algorithm and fractional peeling.

Original: Our work also addsvalue via a detailed comparison of the practical performance of existing algorithms and the newalgorithm.
Revised: Our work also addsvalue via a detailed comparison of the practical performance of existing algorithms and the newalgorithm.

Original: A few limitations remain for the paper.
Revised: A few limitations remain for the paper.

Original: First, reducing the p n dependency in Theorem 5.6 is important, even if fractional peeling shows strong experimental bounds.
Revised: First, reducing the p n dependency in Theorem 5.6 is important, even if fractional peeling shows strong experimental bounds.

Original: Finally, we needto evaluate empirically the quality of the decomposition achieved by algorithms (with and withoutfractional peeling), and also algorithms for approximating at least k DSG and related problems viathe decomposition. 
Revised: Finally, we needto evaluate empirically the quality of the decomposition achieved by algorithms (with and withoutfractional peeling), and also algorithms for approximating at least k DSG and related problems viathe decomposition. Checklist For all authors...

Revised: (a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’s

Revised: contributions and scope?

Revised: [Yes]

Revised: (b) Did you describe the limitations of your work?

Revised: [Yes]

Revised: In the experimental discussion,

Revised: we describe the limitations of different algorithms (including ours) on the datasets

Revised: (c) Did you discuss any potential negative societal impacts of your work?

Revised: [N/A] Work is

Revised: mainly theoretical with an applied ﬂavor and doesn’t involve inherent potential soci-

Revised: etal impacts.

Revised: (d) Have you read the ethics review guidelines and ensured that your paper conforms to

Revised: them?

Revised: [Yes]

Revised: If you are including theoretical results...

Revised: (a) Did you state the full set of assumptions of all theoretical results?

Revised: [Yes]

Revised: (b) Did you include complete proofs of all theoretical results?

Revised: [Yes] All proofs are included in Appendix

Revised: If you ran experiments...

Revised: (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?

Revised: [Yes] Code is insupplemental section, Data available online (not our data to include)

Revised: (b) Did you specify all the training details (e.g., data splits, hyperparameters, how theywere chosen)?

Revised: [Yes]

Revised: (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?

Revised: [N/A]

Revised: The algorithm isn’t randomized, no random seed orerror plots applicable.(d)

Revised: Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)?

Revised: [Yes]

Revised: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...

Revised: (a) If your work uses existing assets, did you cite the creators?

Revised: [Yes] Code for Greedy++

Revised: was cited appropriately together with used libraries.

Revised: All the sources data were also

Revised: cited.

Revised: (b) Did you mention the license of the assets?

Revised: [Yes]

Revised: (c) Did you include any new assets either in the supplemental material or as a URL?

Revised: [Yes]

Revised: (d) Did you discuss whether and how consent was obtained from people whose data

Revised: you’re using/curating?

Revised: [N/A]

Revised: No personal data of people was used

Revised: (e) Did you discuss whether the data you are using/curating contains personally identiﬁ-

Revised: able information or offensive content?

Revised: [N/A] Personal data is not used

Revised: If you used crowdsourcing or conducted research with human subjects...

Revised: (a) Did you include the full text of instructions given to participants and screenshots, if

Revised: applicable?

Revised: [N/A] Crowdsourcing is not used

Revised: (b) Did you describe any potential participant risks, with links to Institutional Review

Revised: Board (IRB) approvals, if applicable?

Revised: [N/A] Crowdsourcing is not used

Revised: (c) Did you include the estimated hourly wage paid to participants and the total amount

Revised: spent on participant compensation?

Revised: [N/A] Crowdsourcing is not used