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Inapproximability of Treewidth and Related Problems (Extended Abstract) ∗ Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence (IJCAI 2015) Inapproximability of Treewidth and Related Problems (Extended Abstract) ∗ Yu (Ledell) Wu y, Per Austrin z, Toniann Pitassi and David Liu Department of Computer Science, University of Toronto, Ontario, Canada fwuyu,austrin,toni,[email protected] Abstract the distribution that allows one to express it more succinctly. A graphical model represents such a joint probability distribu- Graphical models, such as Bayesian Networks and tion by a graph where the vertices represent the random vari- Markov networks play an important role in artifi- ables, and the dependences are modeled by the graph struc- cial intelligence and machine learning. Inference is ture. Associated with each vertex of the graph is a conditional a central problem to be solved on these networks. probability table, which specifies the conditional probabili- This, and other problems on these graph models ties of this random variable, conditioned on its neighboring are often known to be hard to solve in general, vertices. The two most common types of graphical models but tractable on graphs with bounded Treewidth. are Bayesian networks (also called belief networks), where Therefore, finding or approximating the Treewidth the underlying graph is directed, and Markov networks (also of a graph is a fundamental problem related to in- called Markov random fields), where the underlying graph is ference in graphical models. In this paper, we study undirected. The most basic problem in graphical models is the approximability of a number of graph problems: the inference problem, which is the problem of computing Treewidth and Pathwidth of graphs, Minimum Fill- the posterior marginal distribution of a variable at some ver- In, and a variety of different graph layout problems tex. Unfortunately, inference in general is well-known to be such as Minimum Cut Linear Arrangement. We NP-hard to compute exactly as well as to approximate [Roth, show that, assuming Small Set Expansion Conjec- 1996]. ture, all of these problems are NP-hard to approx- Despite this intractability, an important class of bounded imate to within any constant factor in polynomial Treewidth instances of probabilistic inference has been iden- time. tified and shown to be exactly computable in polynomial This paper is an extended abstract of the Journal of time. The Treewidth of a graph [Robertson and Seymour, Artificial Intelligence Research [Wu et al., 2014] 1984] is a fundamental parameter of a graph that measures how close the graph is to being a tree. Treewidth is very 1 Introduction closely related to the other notions in machine learning such as Branch-width, Clique-width and Elimination-width (for an Graphical models provide a computational framework for ef- overview of Treewidth and related notions, see [Bodlaender ficiently manipulating probability distributions over high di- et al., 1995]). On graphs with small Treewidth and where mensional spaces, often involving hundreds of thousands of the tree decomposition is known, a dynamic programming al- variables. This framework has found applications in an enor- gorithm yields a polynomial-time algorithm. Particular al- mous range of domains including: medical and fault diag- gorithms for probabilistic inference on bounded Treewidth nosis, image understanding, speech recognition, web search, graphs are the junction-tree method, variable elimination and coding theory, and statistical physics [Koller and Friedman, clique trees (e.g. see [Koller and Friedman, 2009], ch. 9, 10). 2009]. A graphical model is an efficient representation of a The same ideas also yield polynomial-time algorithms and joint distribution over some set of n random variables. Even often even linear time algorithms for small Treewidth in- if the random variables are binary, it is well known that an ar- n stances for an astonishing variety of other NP-hard prob- bitrary joint distribution requires the specification of 2 prob- lems, including: satisfiability, counting satisfying assign- abilities. Luckily, in the real world, there is often structure in ments, constraint satisfaction, vertex cover, maximum inde- ∗This paper is an extended abstract of the Journal of Artificial pendent set, Hamiltonian circuit, matrix decomposition, and Intelligence Research [Wu et al., 2014]. Research supported by more generally all problems definable in monadic second- NSERC. order logic. (See the excellent survey [Bodlaender, 2005] for yPresent Affiliation: Facebook AI Research, 770 Broadway, New motivation, including theoretical as well as practical applica- York, USA. tions of Treewidth.) One catch is that for all of these prob- zPresent Affiliation: School of Computer Science and Commu- lems, the algorithm must begin by finding a tree decompo- nication, KTH Royal Institute of Technology, Stockholm, Sweden sition, and then use the decomposition to solve the problem. 4222 Given the tree decomposition, the algorithm is typically ex- optimal lower bound, since there is a simple algorithm that 7 ponential in the width of the underlying tree decomposition. achieves an approximation ratio of 8 . Despite this success, Thus there is a need for efficient algorithms to actually com- for many important problems, such as Vertex Cover, Max- pute the Treewidth of a given graph, and to find tree decom- Cut, and Kernel Clustering, the hardness of approximation positions with optimal or close to optimal width. results obtained through the PCP theorem have not matched Unfortunately, while there are many good heuristics for the best approximation algorithms known. finding a good tree decomposition, it is NP-hard in gen- The formulation of the Unique Games Conjecture (UGC) eral to determine the Treewidth of a graph [Arnborg et al., due to Khot [Khot, 2002] was intended to clarify the ap- 1987]. However, Bodlaender et al. [Bodlaender et al., proximability of many optimization problems. The conjec- 1995] obtained an O(log n) factor approximation algorithm ture postulates that the problem of determining the value of for Treewidth. In fact, they actually show that if there is a a certain type of game, known as a unique game, is NP-hard. factor c approximation algorithm for vertex separator, then Under UGC, many of the known algorithms in approximation there is an O(c) approximation algorithm for Treewidth. And are proven to be tight (for an excellent survey on this topic, if there is a factor b approximation algorithm for Treewidth see [Khot and Vishnoi, 2005]). For instance, under the UGC, then there is an O(b log n) approximation algorithm for the the Vertex Cover problem is NP-hard to approximate within related Pathwidth problem. The best currentlyp known approx- a factor of 2 − , for any > 0 [Khot and Regev, 2008]. imation factor for vertex separator is O( log n) [Feige et al., Perhaps most strikingly, Raghavendra [Raghavendra, 2008] 2005] and thus the best algorithm for Treewidth finds a tree proved that under the UGC, the semi-definite programming p (SDP) approximation algorithm for a large class of constraint decomposition that is withinp an O( log n) factor of the opti- mal width, and an O(( log n)(log n)) factor approximation satisfaction problems (CSP) are essentially the best one can algorithm for Pathwidth. hope for. Thus, the UGC has become the central open prob- It is a longstanding open question whether or not there is a lem in inapproximability and encapsulates the barrier of de- constant factor approximation algorithm for Treewidth. Such signing better polynomial time approximation algorithms for an algorithm would lead to faster algorithms to find good tree- a large class of problems. decompositions for all of the problems mentioned above. The Despite this tremendous progress, still there remain im- current best known algorithm that achieves a constant factor portant yet stubborn problems such as Treewidth, Balanced approximation for Treewidth runs in time 2O(w)O(n), where Separator, Minimum Linear Arrangement (MLA), and many w is the Treewidth of the underlying graph, and achieves a other graph layout problems whose approximation status re- factor 5 approximation [Bodlaender, 2007]. Similarly, the ap- mains unresolved even assuming the UGC. In the work of proximability of many related graph layout problems is also Raghavendra and Steurer [Raghavendra and Steurer, 2010], unresolved, including Minimum Cut Linear Arrangement and the Small Set Expansion (SSE) Conjecture was introduced, Interval Graph Completion. and it was shown that it implies the UGC, and that the In this paper, we make an important step to resolve this SSE Conjecture follows if one assumes that the UGC is problem by showing that Treewidth, Pathwidth, and a host true for somewhat expanding graphs. In follow-up work by of related graph layout problems are hard to approximate to Raghavendra et al. [Raghavendra et al., 2012], it was shown within any constant factor, under the Small Set Expansion that the SSE Conjecture is in fact equivalent to the UGC on (SSE) conjecture [Raghavendra and Steurer, 2010]. somewhat expanding graphs, and that the SSE Conjecture im- The SSE conjecture is a strengthened version of the con- plies NP-hardness of approximation for balanced separator jecture that P is different from NP and warrants some ex- and MLA. In this light, the Small Set Expansion conjecture planation. In the next subsection (Section 1.1), we explain serves as a natural unified conjecture that yields all of the im- the SSE conjecture, and how it relates to the P versus NP plications of UGC and also hardness for expansion-like prob- question and to related conjectures. We then state our main lems that could not be resolved with the UGC. hardness results for Treewidth, Pathwidth and graph layout Our main contribution in this paper is to prove that a wide problems (Sections 1.2, and 1.4), and discuss related results range of other graph layout problems are SSE-hard to approx- in Section 1.5. imate to within any constant factor. For these problems, no evidence of hardness of approximation was known prior to 1.1 The Small Set Expansion Conjecture our results.
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