Beyond Hypertree Width: Decomposition Methods Without Decompositions

Chen, Hubie; Dalmau, Víctor

doi:10.1007/11564751_15

Cited by 49 publications

(86 citation statements)

References 21 publications

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“…Previously, bounded hypertree width was the most general such property. Answering an open question raised in [Chen and Dalmau 2005;Cohen et al 2008;Gottlob et al 2005;Grohe 2007], we have identified a new class of polynomial-time solvable CSP instances: instances having bounded fractional edge cover number. This result suggests the definition of fractional hypertree width, which is always at most as large as the hypertree width (and in some cases much smaller).…”

Section: Discussionmentioning

confidence: 99%

“…For a class H of hypergraphs, we let Csp(H) be the class of all instances whose hypergraph is contained in H. The central question is for which classes H of hypergraphs the problem Csp(H) is tractable. Most recently, this question has been studied in [Chen and Dalmau 2005;Cohen et al 2008;Marx 2011;Marx 2010b;Gottlob et al 2005]. It is worth pointing out that the corresponding question for the graphs (instead of hypergraphs) of instances, in which two variables are incident if they appear together in a constraint, has been completely answered in [Grohe 2007;Grohe et al 2001] (under the complexity theoretic assumption FPT = W[1]): For a class G of graphs, the corresponding problem Csp(G) is in polynomial time if and only if G has bounded tree width.…”

Section: Introductionmentioning

confidence: 99%

“…On classes of bounded hyperedge size, bounded hypertree width coincides with bounded tree width, but in general it does not. It has been asked in [Chen and Dalmau 2005;Cohen et al 2008;Gottlob et al 2005;Grohe 2007] whether there are classes H of unbounded hypertree width such that Csp(H) ∈ PTIME. We give an affirmative answer to this question.…”

Section: Introductionmentioning

confidence: 99%

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Constraint solving via fractional edge covers

Grohe

Marx

2006

Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm - SODA '06

118

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Many important combinatorial problems can be modeled as constraint satisfaction problems. Hence identifying polynomial-time solvable classes of constraint satisfaction problems has received a lot of attention. In this paper, we are interested in structural properties that can make the problem tractable. So far, the largest structural class that is known to be polynomial-time solvable is the class of bounded hypertree width instances introduced by Gottlob et al. [2002]. Here we identify a new class of polynomial-time solvable instances: those having bounded fractional edge cover number.Combining hypertree width and fractional edge cover number, we then introduce the notion of fractional hypertree width. We prove that constraint satisfaction problems with bounded fractional hypertree width can be solved in polynomial time (provided that a the tree decomposition is given in the input). Together with a recent approximation algorithm for finding such decompositions [Marx 2010a], it follows that bounded fractional hypertree width is now the most general known structural property that guarantees polynomialtime solvability.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Constraint solving via fractional edge covers

Grohe

Marx

2006

Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm - SODA '06

118

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“…Generalized hypertreewidth seems the natural counterpart of the tree decomposition method over unbounded-arity classes, and in fact CSP(A, −) is solvable in polynomial time if A has bounded hypertree-width modulo homomorphic equivalence [15]. However, it is known that generalized hypertree-width does not characterize all classes of structures where CSP(A, −) is solvable in polynomial time.…”

Section: Decision Problemmentioning

confidence: 99%

Treewidth and Hypertree Width

Gottlob¹,

Greco²,

Scarcello³

2014

Tractability

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The chapter covers methods for identifying islands of tractability for NP-hard combinatorial problems by exploiting suitable properties of their graphical structure. Acyclic structures are considered, as well as nearly acyclic ones identified by means of so-called structural decomposition methods. In particular, the chapter focuses on the tree decomposition method, which is the most powerful decomposition method for graphs, and on the hypertree decomposition method, which is its natural counterpart for hypergraphs. These problem-decomposition methods give rise to corresponding notions of width of an instance, namely, treewidth and hypertree width. It turns out that many NP-hard problems can be solved efficiently over classes of instances of bounded treewidth or hypertree width: deciding whether a solution exists, computing a solution, and even computing an optimal solution (if some cost function over solutions is specified) are all polynomial-time tasks. Example applications include problems from artificial intelligence, databases, game theory, and combinatorial auctions.

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“…As an example, the degree of acyclicity of a constraint graph, measured using various graph width parameters, plays an important role with respect to the identification of tractable instances -it is known that an instance is solvable in polynomial time if the treewidth of its constraint graph is bounded by a constant [8,9,6,10,5,21]. Interestingly, even though the notion of bounded treewidth is defined with respect to tree decompositions, it is also possible to design algorithms for constraint satisfaction problems of bounded (generalized) hypertree width that do not perform any form of tree decomposition (see e.g., [3]). Other useful structural properties consider the nature of the constraints, such as their so-called functionality, monotonicity, and row convexity [7,24].…”

Section: Introductionmentioning

confidence: 99%

Tradeoffs in the Complexity of Backdoor Detection

Dilkina

Gomes

Sabharwal

Principles and Practice of Constraint Programming – CP 2007

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Abstract. There has been considerable interest in the identification of structural properties of combinatorial problems that lead to efficient algorithms for solving them. Some of these properties are "easily" identifiable, while others are of interest because they capture key aspects of state-of-the-art constraint solvers. In particular, it was recently shown that the problem of identifying a strong Horn-or 2CNF-backdoor can be solved by exploiting equivalence with deletion backdoors, and is NPcomplete. We prove that strong backdoor identification becomes harder than NP (unless NP=coNP) as soon as the inconsequential sounding feature of empty clause detection (present in all modern SAT solvers) is added. More interestingly, in practice such a feature as well as polynomial time constraint propagation mechanisms often lead to much smaller backdoor sets. In fact, despite the worst-case complexity results for strong backdoor detection, we show that Satz-Rand is remarkably good at finding small strong backdoors on a range of experimental domains. Our results suggest that structural notions explored for designing efficient algorithms for combinatorial problems should capture both statically and dynamically identifiable properties.

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Beyond Hypertree Width: Decomposition Methods Without Decompositions

Cited by 49 publications

References 21 publications

Constraint solving via fractional edge covers

Constraint solving via fractional edge covers

Treewidth and Hypertree Width

Tradeoffs in the Complexity of Backdoor Detection

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