arXiv:2105.09675v1 [cs.DS] 20 May 2021 rjc PRH O3669/5-1. KO OPERAH, project aeinntokcnit faDAG a of consists distributi network probability Bayesian joint modelling in for tool dependencies important statistical most the are networks Bayesian Introduction 1 nti rbe,w r ie oa aetsoefunc- score parent local a given are we tion problem, this In ewr iha pia ewr tutr sN-adas NP-hard instances restricted is very structure Bayesian on network a even optimal constructing well, of an task task with inference the this network Moreover, that is NP-hard. networks is Bayesian draw using t the of of of One backs its distributions observations. of the probability under some the variables on remaining infer values may observed one the variables, and network Bayesian a set able olanaDAG a learn to vralvralsi maximal. is variables all over ∗ upre yteDush oshnseencat(DFG), Forschungsgemeinschaft Deutsche the by Supported f etlts ndt cec st er Bayesian a P learn to data. is observed funda- from science A network data in distribution. task probability mental joint vari- a between of ables dependencies that statistical graph acyclic directed represents a is network Bayesian A h ubro aibe and variables of number the t neligudrce rp safrs.In forest. P of a complexity the is revisit that we graph work, property this undirected additional underlying the its fulfills that network ftenme fvariables influence of the number consider the we of Moreover, size. stance oepyprn e ntefia A ntecom- the on P DAG final of the plexity in set parent nonempty one,te ecnoti fcetalgorithms. efficient obtain can we then bounded, otat if contrast, a esle in solved be can which learning network Bayesian unlike algorithm, L ING ING P v OLYTREE EARNING N 2 : stepolmo eriga pia Bayesian optimal an learning of problem the is a esle in solved be can n e fcniiinlpoaiiytbe.Given tables. probability condiditional of set a and N \{ v ( } eso htP that show We . d ,A N, L OLYTREE → n h aiu aetstsz are size set parent maximum the and EARNING ntePrmtrzdCmlxt fPlte Learning Polytree of Complexity Parameterized the On il Grüttemeier Niels 2 ) N d uhta h u fteprn scores parent the of sum the that such 0 | · Abstract ner,kmseiz morawietz}@informatik.uni-marbu komusiewicz, {niegru, o ahvariable each for I 3 | L n O a no has EARNING (1) | · I d hlpsUiesttMrug abr,Germany Marburg, Marburg, Philipps-Universität ie eso ht in that, show We time. | O htmgtrciea receive might that (1) | OLYTREE f OLYTREE I ( | ( d eso that show We . iewhere time ,A N, stettlin- total the is ) [ hceig 1995 Chickering, | · v I , ) | n h akis task the and OLYTREE O vrtevari- the over hita Komusiewicz Christian L L (1) EARN EARN -time n n.A ons. is - - he ] - . inare tion re aebe sd o xml,i mg-emnainfo image-segmentation in data example, microscopy for used, been have trees esfrBysa ewrsflligsm spe- exten- studied some been fulfilling prob- have networks inference constraints and Bayesian sively structural learning for cific the lems networks, Bayesian aawm n zie,2021 Szeider, and Ramaswamy 2020a; Komusiewicz, and Grüttemeier 2015; Parviainen, and Korhonen n.I te od,eeycnetdcmoeti di- in a performed is be can component trees inference on connected time and every polynomial par- Learning one words, in-tree. most other rected at In has vertex ent. every where network Bayesian s.A datg fpltesi htteifrnetask inference the them that on time is for- polynomial polytrees a in of is poly- performed be graph advantage A can undirected An underlying of whose proposed. generalization DAG est. been a a has is problem, other tree polytrees one this called most overcome power at branchings To modeling on only their depend variable. in may limited variable every very since however, are, Trees eae Work. Related tal. et u n s,2002 Hsu, and Guo loihsfrP for algorithms steoealiptsz.Tu,ternigtm fteealg fixed these every of for time polynomial running the is Thus, rithms size. input overall the is flann notmlplte tutr rmprn score parent from NP-hard structure remains polytree however, optimal an learning of feeyprn e ihsrcl oiiesoehssize has score positive strictly with 2 set most parent at every if ewe h Phrns fP of NP-hardness the between a etasomdit reb tmost at that it by those tree precisely, among a polytree More into best transformed the considered. be that been algo- can shown are polynomial-time has been that a polytrees trees has learning has to optimally tree of close a problem the learning rithm, that fact the Safaei in a efudin found be can tions ING n fteeris pca ae hthsrcie atten- received has that cases special earliest the of One nlgto h adeso adiggeneral handling of hardness the of light In ol edt consider to need would [ 2015 tal. et tree [ ] er,18;Kroe n avann 2013; Parviainen, and Korhonen 1989; Pearl, 2013 , rt-oc loih o P for algorithm brute-force a , [ agpa 1999 Dasgupta, and ewrs localled also networks, OLYTREE [ Fehri ] P isMorawietz Nils ] where OLYTREE P . n O tal. et ( OLYTREE k [ n ) hwadLu 98 er,1989 Pearl, 1968; Liu, and Chow L | [ I agpa 1999 Dasgupta, stenme fvralsand variables of number the is n EARNING ] 2019 , | oiae ytecontrast the by Motivated . n O rg.de − L (1) ] . 2 EARNING OLYTREE · time ] 2 L . branchings n k EARNING − . sntdb Gaspers by noted As . ∗ 1 [ Gaspers ietdtes Poly- trees. directed OLYTREE ] sN-adeven NP-hard is esuyexact study We . L EARNING h problem the , k rei a is tree A . [ er,1989; Pearl, tal. et dedele- edge L 2015; , EARN and | o- I s, ] - r | . A Our Results. We obtain an algorithm that solves POLY- that has no children is a sink.Given a vertex v, we let Pv := TREE LEARNING in 3n · |I|O(1) time. This is the first algo- {u | (u, v) ∈ A} denote the parent set of v. The skeleton rithm for POLYTREE LEARNING where the running time is of a directed graph (N, A) is the undirected graph (N, E) singly-exponential in the number of variables n. Therunning where {u, v} ∈ E if and only if (u, v) ∈ A or (v,u) ∈ time is a substantial improvement over the brute-force algo- A. A directed acyclic graph is a polytree if its skeleton is a rithm mentioned above, thus positively answering a question forest, that is, the skeleton is acyclic [Dasgupta, 1999]. Asa of Gaspers et al. [2015] on the existence of such algorithms. shorthand, we write P × v := P ×{v} for a vertex v and a We then study whether POLYTREE LEARNING is amenable set P ⊆ N. to polynomial-time data reduction that shrinks the instance I Problem Definition. Given a vertex set N, a family F := if |I| is much bigger than n. Using tools from parameter- {f : 2N\{v} → N | v ∈ N} is a family of local scores ized complexity theory, we show that such a data reduction v 0 for N. Intuitively, fv(P ) is the score that a vertex v obtains algorithm is unlikely. More precisely, we show that (un- if P is its parent set. Given a directed graph D := (N, A) we der standard complexity-theoretic assumptions) there is no A define score(A) := v∈N fv(Pv ). We study the following polynomial-time algorithm that replaces any n-variable in- computational problem. stance I by an equivalent one of size nO(1). In other words, P it seems necessary to keep an exponential number of parent POLYTREE LEARNING sets to compute the optimal polytree. Input: A set of vertices N, local scores F = {fv | v ∈ We then consider a parameter that is potentially much N}, and an integer t ∈ N0. Question smaller than n and determine whether POLYTREE LEARN- : Is there an arc-set A ⊆ N × N such ING can be solved efficiently when this parameter is small. that (N, A) is a polytree and score(A) ≥ t? The parameter d, whichwe callthenumberof dependent vari- Given an instance I := (N, F,t) of POLYTREE LEARN- ables, is the number of variables v for which at least one en- ING, an arc-set A is a solution of I if (N, A) is a try of fv is strictly positive. The parameter essentially counts polytree and score(A) ≥ t. Without loss of general- how many variables might receive a nonemptyparent set in an ity we may assume that fv(∅) = 0 for every v ∈ optimal solution. We show that POLYTREE LEARNING can N [Grüttemeier and Komusiewicz, 2020a]. Throughout this be solved in polynomial time when d is constant but that an work, we let n := |N|. O(1) algorithm with running time g(d) · |I| is unlikely for any Solution Structure and Input Representation. We as- computable function g. Consequently, in order to obtain pos- sume that the local scores F are given in non-zero represen- itive results for the parameter d, one needs to consider further tation. That is, f (P ) is part of the input if it is different restrictions on the structure of the input instance. We make a v from 0. For N = {v1,...,vn}, the local scores F are repre- first step in this direction and consider the case where all par- sented by a two-dimensional array [Q ,Q ,...,Q ], where ent sets with a strictly positive score have size at most p. Us- 1 2 n each Qi is an array containing all triples (fvi (P ), |P |, P ) ing this parameterization, we show that every input instance where f (P ) > 0. The size |F| is defined as the number ωdp O(1) vi can be solved in 2 ·|I| time where ω is the matrix mul- of bits needed to store this two-dimensional array. Given an tiplication constant. With the current-best known value for ω instance I := (N, F,t), we define |I| := n + |F| + log(t). dp O(1) this gives a running time of 5.18 · |I| . We then consider For a vertex v, we call PF (v) := {P ⊆ N \{v} | again data reduction approaches. This time we obtain a posi- fv(P ) > 0} ∪ {∅} the set of potential parents of v. Given tive result: Any instance of POLYTREE LEARNING where p is a yes-instance I := (N, F,t) of POLYTREE LEARNING, constant can be reduced in polynomial time to an equivalent A there exists a solution A such that Pv ∈ PF (v) for ev- O(1) A one of size d . Informally, this means that if the instance ery v ∈ N: If there is a vertex with Pv 6∈ PF (v) has only few dependent variables, the parent sets with strictly we can simply replace its parent set by ∅. The run- positive score are small, and there are many nondependent ning times presentend in this paper will also be measured variables, then we can identify some nondependent variables in the maximum number of potential parent sets δF := that are irrelevant for an optimal polytree representing the in- maxv∈N |PF (v)| [Ordyniak and Szeider, 2013]. put data. We note that this result is tight in the following A tool for designing algorithms for POLYTREE LEARNING sense: Under standard complexity-theoretic assumptions it is is the directed superstructure [Ordyniak and Szeider, 2013] impossible to replace each input instance in polynomial time which is the directed graph SF := (N, AF ) with AF := O(1) by an equivalent one with (d + p) variables. Thus, the {(u, v) | ∃P ∈ PF (v): u ∈ P }. In other words, there is an assumption that p is a constant is necessary. arc (u, v) ∈ AF if and only if u is a potential parent of v. Parameterized Complexity. A problem is in the class XP 2 Preliminaries for a parameter k if it can be solved in |I|g(k) time for some Notation. An undirected graph is a tuple (V, E), where V computable function g. In other words, a problem is in XP is a set of vertices and E ⊆ {{u, v} | u, v ∈ V } is a set of if it can be solved within polynomial time for every fixed k. edges. Given a vertex v ∈ V , we define NG(v) := {u ∈ V | A problem is fixed-parameter tractable (FPT) for a parame- {u, v} ∈ E} as the neighborhood of v. A directed graph is a ter k if it can be solved in g(k) · |I|O(1) time for some com- tuple (N, A), where N is a set of verticesand A ⊆ N ×N is a putable g. If a problem is W[1]-hard for a parameter k, then set of arcs. If (u, v) ∈ A we call u a parent of v and v a child it is assumed to not be fixed-parameter tractable for k. A of u. A vertex that has no parents is a source and a vertex problem kernel is an algorithm that, given an instance I with S and i ∈ [1, |P |] is ′ T [v,P,S,i] := max′ T [v,P,S \ S ,i − 1]+ P p1 p2 p3 p4 p5 S ⊆S max max T [v′, P ′, (S′ ∪{p }) \ (P ′ ∪{v′}), |P ′|]. ′ ′ ′ ′ i v ∈S ∪{pi} P ∈PF (v ) ′ ′ v P ⊆S ∪{pi} Note that the two vertex sets P ∪ (S \ S′) ∪{v} and P ′ ∪ ′ ′ ′ ′ ′ Figure 1: Illustration of an entry T [v,P,S,i] where i = 3. (S ∪{pi}) \ (P ∪{v }) ∪{v } = S ∪{pi} share only the vertex pi. Hence, combining the polytree on these two vertex sets results in a polytree. parameter k computes in polynomial time an equivalent in- If i is equal to zero, then the recurrence is stance I′ with parameter k′ such that |I′| + k′ ≤ h(k) for some computable function h. If h is a polynomial, then the T [v, P, S, 0] := problem admits a polynomial kernel. Strong conditional run- ′ ′ ′ ′ ′ ′ fv(P )+max′ ′ max ′ T [v , P ,S \ (P ∪{v }), |P |]. v ∈S P ∈PF (v ) ning time bounds can also be obtained by assuming the Ex- ′ P ⊆S ponential Time Hypothesis (ETH), a standard conjecture in complexity theory [Impagliazzo et al., 2001]. For a detailed Thus, to determine if I is a yes-instance of POLYTREE introduction into parameterized complexity we refer to the LEARNING, it remains to check if T [v,P,N \ (P ∪ standard textbook [Cygan et al., 2015]. {v}), |P |] ≥ t for some v ∈ N and some P ∈ PF (v). The corresponding polytree can be found via traceback. The cor- rectness proof is straightforward and thus omitted. 3 Parameterization by the Number of Vertices n 2 The dynamic programming table T has 2 · n · δF entries. |S| 2 In this section we study the complexity of POLYTREE Each of these entries can be computed in 2 ·n ·δF . Conse- LEARNING when parameterized by n, the number of vertices. quently, all entries can be computed in n n 2i ·|I|O(1) = n−1 i=0 i Note that there are up to n·2 entries in F and thus, the to- 3n · |I|O(1) time in total. To evaluate if there is some v ∈ N tal input size of an instance of POLYTREE LEARNING might P and some P ∈PF (v) such that T [v,P,N \(P ∪{v}), |P |] ≥ be exponential in n.On the positive side, we show that the t can afterwards be done in O(n · δ ) time. Hence, the total n O(1) F problem can be solved in O(3 · |I| ) time. running time is 3n · |I|O(1). On the negative side, we complement the result above by proving that POLYTREE LEARNING does not admit a poly- A Kernel Lower Bound In this subsection we prove nomial kernel when parameterized by n. In other words, it the following kernel lower bound. The proof is closely is presumably impossible to transform an instance of POLY- related to a similar kernel lower bound for BAYESIAN TREE LEARNING in polynomial time into an equivalent in- NETWORK STRUCTURE LEARNING parameterized stance of size |I| = nO(1). This shows that it is sometimes by n [Grüttemeier and Komusiewicz, 2020a]. necessary to keep an exponential number of parent scores to Theorem 3.2. POLYTREE LEARNING does not admit a poly- compute an optimal polytree. nomial kernel when parameterized by n, unless NP ⊆ coNP/poly. 3.1 An FPT Algorithm Proof. We give a polynomial parameter transforma- Theorem 3.1. POLYTREE LEARNING can be solved in 3n · tion [Bodlaender et al., 2011] from MULTICOLORED |I|O(1) time. INDEPENDENT SET. MULTICOLORED INDEPENDENT SET Proof. Let I := (N, F,t) be an instance of POLYTREE Input LEARNING. We describe a dynamic programming algorithm : An integer k, a graph G = (V, E), and a par- to solve I. We suppose an arbitrary fixed total ordering on the tition (V1,...,Vk) of V such that G[Vi] is a clique for vertices of N. For every P ⊆ N, and every i ∈ [1, |P |], we each i ∈ [1, k]. Question: Does G contain an independent set of denote with pi the ith smallest element of P according to the total ordering. size k? The dynamic programming table T has entries of The parameter for this polynomial parameter transformation type T [v,P,S,i] with v ∈ N, P ∈ PF (v), S ⊆ N \ (P ∪ is the number of vertices not contained in Vk. {v}), and i ∈ [0, |P |]. Given an instance I = (G = (V, E), k) of MULTI- Each entry stores the maximal score of an arc set A of a COLORED INDEPENDENT SET with partition (V1,...,Vk) polytree on {v} ∪ P ∪ S where v has no children, v learns with ℓ := |V | − |Vk|, we construct in polynomial time an exactly the parent set P under A, and foreach j ∈ [i+1, |P |], equivalent instance I′ = (N, F,t) of POLYTREE LEARNING 2
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