The Complexity of Bayesian Network Learning: Revisiting the Superstructure

1 Introduction . Bayesian networks are among the most prominent graphical models for probability distributions . The key feature of Bayesian networks is that they represent conditional dependencies between random variables via a directed acyclic graph ; the vertices of this graph are the variables , and an arc ab means that the distribution of variable b depends on the value of a . One beneficial property of Bayesian networks is that they can be used to infer the distribution of random variables in the network based on the values of the remaining variables . The problem of constructing a Bayesian network with an optimal network structure is NP-hard , and remains NP-hard even on highly restricted instances [ 4 ] . This initial negative result has prompted an extensive investigation of the problem ’ s complexity , with the aim of identifying new tractable fragments as well as the boundaries of its intractability [ 26 , 32 , 27 , 22 , 11 , 6 , 19 ] . The problem— which we simply call BAYESIAN NETWORK STRUCTURE LEARNING ( BNSL ) —can be stated as follows : given a set of V of variables ( represented as vertices ) , a family F of score functions which assign each variable v ∈ V a score based on its parents , and a target value ` , determine if there exists a directed acyclic graph over V that achieves a total score of at least ` 1 . 1Formal definitions are provided in Section 2 . We consider the decision version of BNSL for complexitytheoretic reasons only ; all of the provided algorithms are constructive and can output a network as a witness . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . To obtain a more refined understanding of the complexity of BNSL , past works have analyzed the problem not only in terms of classical complexity but also from the perspective of parameterized complexity [ 9 , 5 ] . In parameterized complexity analysis , the tractability of problems is measured with respect to the input size n and additionally with respect to a specified numerical parameter k. In particular , a problem that is NP-hard in the classical sense may—depending on the parameterization used—be fixed-parameter tractable ( FPT ) , which is the parameterized analogue of polynomial-time tractability and means that a solution can be found in time f ( k ) ·nO ( 1 ) for some computable function f , or W [ 1 ] -hard , which rules out fixed-parameter tractability under standard complexity assumptions . The use of parameterized complexity as a refinement of classical complexity is becoming increasingly common and has been employed not only for BNSL [ 26 , 32 , 27 ] , but also for numerous other problems arising in the context of neural networks and artificial intelligence [ 13 , 41 , 10 , 16 ] . Unfortunately , past complexity-theoretic works have shown that BNSL is a surprisingly difficult problem . In particular , not only is the problem NP-hard , but it remains NP-hard even when asking for the existence of extremely simple networks such as directed paths [ 29 ] and is W [ 1 ] -hard when parameterized by the vertex cover number of the network [ 27 ] . In an effort to circumvent these lower bounds , several works have proposed to instead consider restrictions to the so-called superstructure , which is a graph that , informally speaking , captures all potential dependencies between variables [ 42 , 35 ] . Ordyniak and Szeider [ 32 ] studied the complexity of BNSL when parameterized by the structural properties of the superstructure , and showed that parameterizing by the treewidth [ 36 ] of the superstructure is sufficient to achieve a weaker notion of tractability called XP-tractability . However , they also proved that BNSL remains W [ 1 ] -hard when parameterized by the treewidth of the superstructure [ 32 , Theorem 3 ] . Contribution . Up to now , no “ implicit ” restrictions of the superstructure were known to lead to a fixed-parameter algorithm for BNSL alone . More precisely , the only known fixed-parameter algorithms for the problem require that we place explicit restrictions on either the sought-after network or the parent sets on the input : BNSL is known to be fixed-parameter tractable when parameterized by the number of arcs in the target network [ 22 ] , the treewidth of an “ extended superstructure graph ” which also bounds the maximum number of parents a variable can have [ 26 ] , or the number of parent set candidates plus the treewidth of the superstructure [ 32 ] . Moreover , a closer analysis of the reduction given by Ordyniak and Szeider [ 32 , Theorem 3 ] reveals that BNSL is also W [ 1 ] -hard when parameterized by the treedepth , pathwidth , and even the vertex cover number of the superstructure alone . The vertex cover number is equal to the vertex deletion distance to an edgeless graph , and hence their result essentially rules out the use of the vast majority of graph parameters ; among others , any structural parameter based on vertex deletion distance . As our first conceptual contribution , we show that a different kind of graph parameters—notably , parameters that are based on edge deletion distance—give rise to fixed-parameter algorithms for BNSL in its full generality , without requiring any further explicit restrictions on the target network or parent sets . Our first result in this direction concerns the feedback edge number ( fen ) , which is the minimum number of edges that need to be deleted to achieve acyclicity . In Theorem 3 we show not only that BNSL is fixed-parameter tractable when parameterized by the fen of the superstructure , but also provide a polynomial-time preprocessing algorithm that reduces any instance of BNSL to an equivalent one whose number of variables is linear in the fen ( i.e. , a kernelization [ 9 , 5 ] ) . Since fen is a highly “ restrictive ” parameter—its value can be large even on simple superstructures such as collections of disjoint cycles—we proceed by asking whether it is possible to lift fixedparameter tractability to a more relaxed way of measuring distance to acyclicity . For our second result , we introduce the local feedback edge number ( lfen ) , which intuitively measures the maximum edge deletion distance to acyclicity for cycles intersecting any particular vertex in the superstructure . In Theorem 6 , we show that BNSL is also fixed-parameter tractable when prameterized by lfen ; we also show that this comes at the cost of BNSL not admitting any polynomial-time preprocessing procedure akin to Theorem 3 when parameterized by lfen . We conclude our investigation in the direction of parameters based on edge deletion distance by showing that BNSL parameterized by treecut width [ 28 , 45 , 14 ] , a recently discovered edge-cut based counterpart to treewidth , remains W [ 1 ] -hard ( Theorem 9 ) . An overview of these complexity-theoretic results is provided in Figure 1 . As our second conceptual contribution , we show that BNSL becomes significantly easier when one can use an additive representation of the scores rather than the non-zero representation that was considered in the vast majority of complexity-theoretic works on BNSL to date [ 26 , 32 , 27 , 22 , 11 , 19 ] . The additive representation is inspired by known heuristics for BNSL [ 40 , 39 ] and utilizes a succinct encoding of the score function which assumes that the scores for parent sets can be decomposed into a sum of the scores of individual variables in the parent set ; a discussion and formal definitions are provided in Section 2 . In Theorem 11 , we show that if the additive representation can be used , BNSL becomes fixed-parameter tractable when parameterized by the treewidth of the superstructure ( and hence under every parameterization depicted in Figure 1 ) . Motivated by the empirical usage of the additive representation , we also consider the case where we additionally impose a bound q on the number of parents a vertex can accept ; we show that the result of Theorem 11 also covers this case if q is taken as an additional parameter , and otherwise rule out fixed-parameter tractability using an intricate reduction ( Theorem 12 ) . For our third and final conceptual contribution , we show how our results can be adapted for the emergent problem of POLYTREE LEARNING ( PL ) , a variant of BNSL where we require that the network forms a polytree . The crucial advantage of such networks is that they allow for a more efficient solution of the inference task [ 34 , 23 ] , and the complexity of PL has been studied in several works [ 21 , 19 , 38 ] . We show that all our results for BNSL can be adapted to PL , albeit in some cases it is necessary to perform non-trivial modifications . Furthermore , we observe that unlike BNSL , PL becomes polynomial-time tractable when the additive representation is used ( Observation 16 ) ; this matches the “ naive ” expectation that learning simple networks would be easier than BNSL in its full generality . As our concluding result , we show that this expectation is in fact not always validated : while PL was recently shown to be W [ 1 ] -hard when parameterized by the number of so-called dependent vertices [ 21 ] , in Theorem 17 we prove that BNSL is fixed-parameter tractable under that same parameterization . 2 Preliminaries . For an integer i , we let [ i ] = { 1 , 2 , . . . , i } and [ i ] 0 = [ i ] ∪ { 0 } . We denote by N the set of natural numbers , by N0 the set N ∪ { 0 } . We refer to the handbook by Diestel [ 8 ] for standard terminology on directed as well as undirected graphs . The skeleton ( sometimes called the underlying undirected graph ) of a directed graph ( a digraph ) D = ( V , A ) is the undirected graph G′ = ( V , E ) such that vw ∈ E if vw ∈ A or wv ∈ A . A digraph is a polytree if its skeleton is a forest . When comparing two numerical parameters α , β of graphs , we say that α is more restrictive than β if there exists a function f such that β ( G ) ≤ f ( α ( G ) ) holds for every graph G. We refer to the standard sources for the fundamentals of parameterized complexity , including the definitions of fixed-parameter tractability , parameterized reductions , W [ 1 ] -hardness and treewidth [ 5 , 9 , 31 ] . Problem Definitions . Let V be a set of vertices and F = { fv : 2V \ { v } → N0 | v ∈ V } be a family of local score functions . For a digraph D = ( V , A ) , we define its score as follows : score ( D ) = ∑ v∈V fv ( PD ( v ) ) , where PD ( v ) is the set of vertices of D with an outgoing arc to v ( i.e. , the parent set of v in D ) . We can now formalize our problem of interest [ 32 , 22 ] . BAYESIAN NETWORK STRUCTURE LEARNING ( BNSL ) Input : A set V of vertices , a family F of local score functions , and an integer ` . Question : Does there exist an acyclic digraph D = ( V , A ) such that score ( D ) ≥ ` ? POLYTREE LEARNING ( PL ) is defined analogously , with the only difference that there D is additionally required to be a polytree [ 21 ] . We call D a solution for the given instance . Since both V and F are assumed to be given on the input of our problems , an issue that arises here is that an explicit representation of F would be exponentially larger than |V | . A common way to potentially circumvent this is to use a non-zero representation of the family F , i.e. , where we only store values for fv ( P ) that are different than zero . This model has been used in the vast majority of works studying the complexity of BNSL and PL [ 26 , 32 , 27 , 22 , 19 , 21 ] . Let Γf ( v ) be the collection of candidate parents of v which yield a non-zero score ; formally , Γf ( v ) = { Z | fv ( Z ) 6= 0 } , and the input size |I| of an instance I = ( V , F , ` ) is simply defined as |V |+ ` + ∑ v∈V , P∈Γf ( v ) |P | 2 . A natural way to think about and exploit the structure of inter-variable dependencies of an instance I is to consider its superstructure graph GI = ( V , E ) , where ab ∈ E if a occurs in at least one candidate set in Γf ( b ) ( or vice-versa ) . An example is provided in Figure 2 . Naturally , families of local score functions may be exponentially larger than |V | even when stored using the non-zero representation . In this paper , we also consider a second representation of F which is guaranteed to be polynomial in |V | : in the additive representation , we require that for every vertex v ∈ V and set Q = { q1 , . . . , qm } ⊆ V \ { v } , fv ( Q ) = fv ( { q1 } ) + · · · + fv ( { qm } ) . Hence , each cost function fv can be fully characterized by storing at most |V |-many entries of the form fv ( x ) : = fv ( { x } ) for each x ∈ V \ { v } . To avoid overfitting , one may optionally impose an additional constraint : an upper bound q on the size of any parent set in the solution . While not every family of local score functions admits an additive representation , the additive model is similar in spirit to the models used by some practical algorithms for BNSL . For instance , the algorithms of Scanagatta , de Campos , Corani and Zaffalon [ 40 , 39 ] , which can process BNSL instances with up to thousands of variables , approximate the real score functions by adding up the known score functions for two parts of the parent set and applying a small , logarithmic correction . Both of these algorithms also use the aforementioned bound q for the parent set size . In spite of this connection to practice and the representation ’ s streamlined nature , we are not aware of any prior works that considered the additive representation in complexity-theoretic studies of BNSL and PL . The superstructure graph for the additive representation can be defined in an analogous way as for the non-zero representation : an edge uv simply captures a “ suspected dependencey ’ between variables u and v ( i.e. , one receives a positive score for depending on the other ) . To distinguish between these models , we use BNSL 6=0 , BNSL+ , and BNSL+≤to denote BAYESIAN NETWORK STRUCTURE LEARNING with the non-zero representation , the additive representation , and the additive representation and the parent set size bound q , respectively ( and analogously for PL ) . In our algorithmic results , we will often use G = ( V , E ) to denote the superstructure graph of the input instance I . Without any loss of generality , we will also assume that G is connected . Graph Parameters Based on Edge Cuts . Traditionally , the bulk of graph-theoretic research on structural parameters has focused on parameters that guarantee the existence of small vertex separators in the graph ; these are inherently tied to the theory of graph minors [ 37 , 36 ] and the vertex deletion 2We remark that the non-zero representation could be strengthened even further by omitting each parent set Z of v which contains a proper subset Z′ such that fv ( Z ) ≤ fv ( Z′ ) . This preprocessing step , however , does not have an impact on any of the results presented in this paper . distance . This approach gives rise not only to the classical notion of treewidth , but also to its well-known restrictions and refinements such as pathwidth [ 37 ] , treedepth [ 30 ] and the vertex cover number [ 12 , 25 ] . The vertex cover number is the most restrictive parameter in this hierarchy . However , there are numerous problems of interest that remain intractable even when parameterized by the vertex cover number . A recent approach developed for attacking such problems has been to consider parameters that guarantee the existence of small edge cuts in the graph ; these are typically based on the edge deletion distance or , more broadly , tied to the theory of graph immersions [ 45 , 28 ] . The parameter of choice for the latter is treecut width ( tcw ) [ 45 , 28 , 14 , 15 ] , a counterpart to treewidth which has been successfully used to tackle some problems that remained intractable when parameterized by the vertex cover number [ 17 ] . On the other hand , the by far most prominent parameter based on edge deletion distance is the feedback edge number of a connected graph G = ( V , E ) , which is the minimum cardinality of a set F ⊆ E of edges ( called the feedback edge set ) such that G− F is acyclic . The feedback edge number can be computed in quadratic time and has primarily been used to obtain fixed-parameter algorithms and polynomial kernels for problems where other parameterizations failed [ 17 , 2 , 1 , 44 ] . Up to now , these were the only two edge-cut based graph parameters that have been considered in the broader context of algorithm design . This situation could be seen as rather unstisfactory in view of the large gap between the complexity of the richer class of graphs of bounded treecut width , and the significantly simpler class of graphs of bounded feedback edge number—for instance , the latter class is not even closed under disjoint union . Here , we propose a new parameter that lies “ between ” the feedback edge number and treecut width , and which can be seen as a localized relaxation of the feedback edge number : instead of measuring the total size of the feedback edge set , it only measures how many feedback edges can “ locally interfere with ” any particular part of the graph . Formally , for a connected graph G = ( V , E ) and a spanning tree T of G , let the local feedback edge set at v ∈ V be ETloc ( v ) = { uw ∈ E \ E ( T ) | the unique path between u and w in T contains v } . The local feedback edge number of ( G , T ) ( denoted lfen ( G , T ) ) is then equal to maxv∈V |ETloc ( v ) | , and the local feedback edge number of G is simply the smallest local feedback edge number among all possible spanning trees of G , i.e. , lfen ( G ) = minT is a spanning tree ofG lfen ( G , T ) . It is not difficult to show that the local feedback edge number is “ sandwiched ” between the feedback edge number and treecut width . We also show that computing it is FPT . Proposition 1 . For every graph G , tcw ( G ) ≤ lfen ( G ) + 1 and lfen ( G ) ≤ fen ( G ) . Theorem 2 . The problem of determining whether lfen ( G ) ≤ k for an input graph G parameterized by an integer k is fixed-parameter tractable . Moreover , if the answer is positive , we may also output a spanning tree T such that lfen ( G , T ) ≤ k as a witness . 3 Solving BNSL 6=0 with Parameters Based on Edge Cuts . In this section we provide tractability and lower-bound results for BNSL 6=0 from the viewpoint of superstructure parameters based on edge cuts . Together with the previous lower bound that rules out fixed-parameter algorithms based on all vertex-separator parameters [ 32 , Theorem 3 ] , the results presented here provide a comprehensive picture of the complexity of BNSL 6=0 with respect to superstructure parameterizations . Using the Feedback Edge Number for BNSL 6=0 . We say that two instances I , I ′ of BNSL are equivalent if ( 1 ) they are either both Yes-instances or both No-instances , and furthermore ( 2 ) a solution to one instance can be transformed into a solution to the other instance in polynomial time . Our aim here is to prove the following theorem : Theorem 3 . There is an algorithm which takes as input an instance I of BNSL 6=0 whose superstructure has fen k , runs in time O ( |I|2 ) , and outputs an equivalent instance I ′ = ( V ′ , F ′ , ` ′ ) of BNSL 6=0 such that |V ′| ≤ 16k . In parameterized complexity theory , such data reduction algorithms with performance guarantees are called kernelization algorithms [ 9 , 5 ] . These may be applied as a polynomial-time preprocessing step before , e.g. , more computationally expensive methods are used . The fixed-parameter tractability of BNSL 6=0 when parameterized by the fen of the superstructure follows as an immediate corollary of Theorem 3 ( one may solve I by , e.g. , exhaustively looping over all possible DAGs on V ′ via a brute-force procedure ) . We also note that even though the number of variables of the output instance is polynomial in the parameter k , the instance I ′ need not have size polynomial in k. We begin our path towards a proof of Theorem 3 by computing a feedback edge set EF of G of size k in timeO ( |I|2 ) by , e.g. , Prim ’ s algorithm . Let T be the spanning tree ofG , EF = E ( G ) \E ( T ) . The algorithm will proceed by the recursive application of certain reduction rules , which are polynomialtime operations that alter ( “ simplify ” ) the input instance in a certain way . A reduction rule is safe if it outputs an instance which is equivalent to the input instance . We start by describing a rule that will be used to prune T until all leaves are incident to at least one edge in EF . Reduction Rule 1 . Let v ∈ V be a vertex and let Q be the set of neighbors of v with degree 1 in G. We construct a new instance I ′ = ( V ′ , F ′ , ` ) by setting : 1 . V ′ : = V \ Q ; 2 . Γf ′ ( v ) : = { ∅ } ∪ { ( P \ Q ) | P ∈ Γf ( v ) } ; 3. for all w ∈ V ′ \ { v } , f ′w = fw ; 4. for every P ′ ∈ Γf ′ ( v ) : f ′v ( P ′ ) : = max P : P\Q=P ′ ( fv ( P ) + ∑ vin∈P∩Q fvin ( ∅ ) + ∑ vout∈Q\P max ( fvout ( ∅ ) , fvout ( v ) ) ) . Lemma 4 . Reduction Rule 1 is safe . Observe that the superstructure graph G′ obtained after applying one step of Reduction Rule 1 is simply G−Q ; after its exhaustive application we obtain an instance I such that all the leaves of the tree T are endpoints of EF . Our next step is to get rid of long paths in G whose internal vertices have degree 2 . We note that this step is more complicated than in typical kernelization results using feedback edge set as the parameter , since a directed path Q in G can serve multiple “ roles ” in a hypothetical solution D and our reduction gadget needs to account for all of these . Intuitively , Q may or may not appear as a directed path in D ( which impacts what other arcs can be used in D due to acyclicity ) , and in addition the total score achieved by D on the internal vertices of Q needs to be preserved while taking into account whether the endpoints of Q have a neighbor in the path or not . Because of this ( and unlike in many other kernelization results of this kind [ 17 , 43 , 15 ] ) , we will not be replacing Q merely by a shorter path , but by a more involved gadget . An illustration is provided in Figure 3 . Reduction Rule 2 . Let a , b1 , . . . , bm , c be a path in G such that for each i ∈ [ m ] , bi has degree precisely 2 , and let P = { b1 , . . . , bm } . We construct a new instance I ′ = ( V ′ , F ′ , ` ) as follows : 1 . V ′ : = V ∪ { b } \ { b2 ... bm−1 } ; 2 . Γf ′ ( b ) = { B ∪ { b1 , bm } |B ⊆ { a , c } } where f ′b ( B ∪ { b1 , bm } ) is equal to the maximum score that can be achieved by P if B are used as parents ; 3 . The scores for a and c are obtained from F by simply adding b to any parent set containing either b1 or bm ; 4 . Γf ′ ( b1 ) contains only { a , b , bm } with score equal to the maximum score that can be achieved by P if a is used as a parent but there is no path from a to bm via P ; 5 . Γf ′ ( bm ) contains only { c , b , b1 } with score equal to the maximum score that can be achieved by P if c is used as a parent but there is no path from c to b1 via P ; 6. for all w ∈ V ′ \ { a , b1 , b , bm , c } , f ′w = fw . Lemma 5 . Reduction Rule 2 is safe . Proof of Theorem 3 . We begin by exhaustively applying Reduction Rule 1 on an instance whose superstructure graph has a feedback edge set of size k , which results in an instance with the same feedback edge set but whose spanning tree T has at most 2k leaves . It follows that there are at most 2k vertices with a degree greater than 2 in T . Let us now “ mark ” all the vertices that either are endpoints of the edges inEF or have a degree greater then 2 in T ; the total number of marked vertices is upper-bounded by 4k . We now proceed to the exhaustive application of Reduction Rule 2 , which will only be triggered for sufficiently long paths in T that connect two marked vertices but contain no marked vertices on its internal vertices ; there are at most 4k such paths due to the tree structure of T . Reduction Rule 2 will replace each such path with a set of 3 vertices , and therefore after its exhaustive application we obtain an equivalent instance with at most 4k + 4k · 3 = 16k vertices , as desired . Correctness follows from the safeness of Reduction Rules 1 , 2 , and the runtime bound follows by observing that the total number of applications of each rule as well as the runtime of each rule are upper-bounded by a linear function of the input size . As an immediate corollary of Theorem 3 , we can apply a standard brute-force branching procedure [ 33 ] to solve BNSL in time nO ( 1 ) + 2O ( k ) . Fixed-Parameter Tractability of BNSL 6=0 using the Local Feedback Edge Number . Our aim here will be to lift the fixed-parameter tractability of BNSL 6=0 established by Theorem 3 by relaxing the parameterization to lfen . In particular , we will prove : Theorem 6 . BNSL 6=0 is fixed-parameter tractable when parameterized by the local feedback edge number of the superstructure . Since fen is a more restrictive parameter than lfen , this results in a strictly larger class of instances being identified as tractable . However , the means we will use to establish Theorem 6 will be fundamentally different : we will not use a polynomial-time data reduction algorithm as the one provided in Theorem 3 , but instead apply a dynamic programming approach . Since the kernels constructed by Theorem 3 contain only polynomially-many variables w.r.t . fen , that result is incomparable to Theorem 6 . In fact one can use standard techniques to prove that , under well-established complexity assumptions , a data reduction result such as the one provided in Theorem 3 can not exist for lfen . Theorem 7 . Unless NP ⊆ co-NP/poly , there is no polynomial-time algorithm which takes as input an instance I of BNSL 6=0 whose superstructure has lfen k and outputs an equivalent instance I ′ = ( V ′ , F ′ , ` ′ ) of BNSL 6=0 such that |V ′| ∈ kO ( 1 ) . In particular , BNSL 6=0 does not admit a polynomial kernel when parameterized by lfen . Towards proving Theorem 6 , assume that we are given an instance I = ( V , F , ` ) of BNSL 6=0 with connected superstructure graph G = ( V , E ) . Let T be a fixed rooted spanning tree of G such that lfen ( G , T ) = lfen ( G ) = k , denote the root by r. For v ∈ V ( T ) , let Tv be the subtree of T rooted at v , let Vv = V ( Tv ) , and let V̄v = NG ( Vv ) ∪ Vv . We define the boundary δ ( v ) of v to be the set of endpoints of all edges in G with precisely one endpoint in Vv ( observe that the boundary can never have a size of 1 ) . Notice that |δ ( v ) | ≤ 2k + 2 . We can now proceed to a definition of the records that will be used in our dynamic program . Intuitively , these records will be computed in a leaf-to-root fashion and will store at each vertex v information about the best score that can be achieved by a partial solution that intersects the subtree rooted at v. Let R be a binary relation on δ ( v ) and s an integer . For s ∈ Z , we say that ( R : s ) is a record for a vertex v if and only if there exists a DAG D on V̄v such that ( 1 ) w ∈ Vv for each arc uw ∈ A ( D ) , ( 2 ) ab ∈ R if there exists an a-b path in D , and ( 3 ) s is the total score achieved by D on vertices in Vv . The records ( R , s ) where s is maximal for fixed R are called valid . Denote the set of all valid records for v byR ( v ) , and note that |R ( v ) | ≤ 2O ( k2 ) . Observe that if vi is a closed child of v , thenR ( vi ) consists of precisely two valid records : one for R = ∅ and one for R = { vvi } . Moreover , the root r of T has only a single valid record ( ∅ : sI ) , where sI is the maximum score that can be achieved by a solution in I . The following lemma lies at the heart of our result and shows how we can compute our records in a leaf-to-root fashion along T . Lemma 8 . Let v ∈ V ( G ) have m children in T where m > 0 , and assume we have computedR ( vi ) for each child vi of v. ThenR ( v ) can be computed in time at most m · |Γf ( v ) | · 2O ( k 3 ) . To prove Theorem 6 , we start by invoking Theorem 2 to obtain a spanning tree T and then compute the recordsR ( v ) for each leaf of T via exhaustive branching . We then apply Lemma 8 to propagate our record sets towards the root r of T ; once we obtainR ( r ) , we can output a solution in constant time . The runtime of the dynamic programming procedure used in the proof of Theorem 6 is upper-bounded by |I|3 · 2O ( k3 ) . For our final result for this section , recall that lfen lies between fen and treecut width in the parameter hierarchy ( see Proposition 1 ) . Since BNSL 6=0 is FPT when parameterized by lfen , the next step would be to ask whether this tractability result can be lifted to treecut width . Below , we answer this question negatively via a reduction from the W [ 1 ] -hard MULTICOLORED CLIQUE problem [ 9 , 5 ] . Theorem 9 . BNSL 6=0is W [ 1 ] -hard when parameterized by the treecut width of the superstructure .

Summary: Summary.  The paper is about the parameterized complexity of Bayesian Network Structure Learning. In particular, it is focused to analyze and develop previous contributions on superstructure of the input. The author/s show/s that changing the parametrization yields to achieve fixed-parameter tractability.  The paper also anallyzes how the complexity of structural learning of bayesian networks depends on the input representation, with specific reference to non-zero representation, and additive representation. Finally the paper shows how to extend the main contirbutions presented to the problem of Polytree Learning.