Constraint Solving via Fractional Edge Covers

Grohe, Martin; Marx, Dániel

doi:10.1145/2636918

Cited by 138 publications

(220 citation statements)

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“…When it comes to defining structurally restricted problems that are tractable, one is typically interested in certain parameters of these (hyper)graphs such as tree-width, fractional hypertree width [23], or submodular width [34]. It is, for instance, known that any finite-domain CSP instance I with primal graph G = (V, E) can be solved in ||I|| O(tw(G)) time [14] where tw(G) denotes the treewidth of G, and it can be solved in ||I|| O(fhw(H)) time [23] where fhw(H) denotes the fractional hypertree width of H. Since these results rely on the domains being finite, we restrict ourselves to finite-domain CSPs throughout this section.…”

Section: Structurally Restricted Cspsmentioning

confidence: 99%

“…It is, for instance, known that any finite-domain CSP instance I with primal graph G = (V, E) can be solved in ||I|| O(tw(G)) time [14] where tw(G) denotes the treewidth of G, and it can be solved in ||I|| O(fhw(H)) time [23] where fhw(H) denotes the fractional hypertree width of H. Since these results rely on the domains being finite, we restrict ourselves to finite-domain CSPs throughout this section. Now note that if given a finite constraint language Γ, then the instances of CSP(Γ) are recursively enumerable and CSP(Γ) is in NP.…”

Section: Structurally Restricted Cspsmentioning

confidence: 99%

“…To see this simply note that if the tree-width or fractional hypertree width is restricted by k then such CSP instances can be solved in ||I|| O(k) time [14,23]. If the width parameter under consideration is polynomial-time computable, then we have property P3 (via Lemma 3), too, and conclude that NP-intermediate fragments exist.…”

Section: Structurally Restricted Cspsmentioning

confidence: 99%

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Constructing NP-intermediate problems by blowing holes with parameters of various properties

Jönsson

Lagerkvist

Nordh

2015

Theoretical Computer Science

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The search for natural NP-intermediate problems is one of the holy grails within computational complexity. Ladner's original diagonalization technique for generating NP-intermediate problems, blowing holes, has a serious shortcoming: it creates problems with a highly artificial structure by arbitrarily removing certain problem instances. In this article we limit this problem by generalizing Ladner's method to use parameters with various characteristics. This allows one to define more fine-grained parameters, resulting in NP-intermediate problems where we only blow holes in a controlled subset of the problem. We begin by fully characterizing the problems that admit NP-intermediate subproblems for a broad and natural class of parameterizations, and extend the result further such that structural CSP restrictions based on parameters that are hard to compute (such as tree-width) are covered, thereby generalizing a result by Grohe. For studying certain classes of problems, including CSPs parameterized by constraint languages, we consider more powerful parameterizations. First, we identify a new method for obtaining constraint languages Γ such that CSP(Γ) are NP-intermediate. The sets Γ can have very different properties compared to previous constructions (by, for instance, Bodirsky & Grohe) and provides insights into the algebraic approach for studying the complexity of infinite-domain CSPs. Second, we prove that the propositional abduction problem parameterized by constraint languages admits NP-intermediate problems.This settles an open question posed by Nordh & Zanuttini.

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Section: Structurally Restricted Cspsmentioning

confidence: 99%

Section: Structurally Restricted Cspsmentioning

confidence: 99%

Section: Structurally Restricted Cspsmentioning

confidence: 99%

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Constructing NP-intermediate problems by blowing holes with parameters of various properties

Jönsson

Lagerkvist

Nordh

2015

Theoretical Computer Science

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“…As an example of a different set of useful "resources" (subproblems), we mention a powerful extension of hypertree decompositions, called fractional hypertree decompositions [52]. According to this notion, the resources are the width-k fractional covers of hypergraph vertices, instead of the integral covers that characterize hypertree decompositions (where each hyperedge counts 1).…”

Section: Beyond (Hyper)tree Decompositionsmentioning

confidence: 99%

“…However, it is known that generalized hypertree-width does not characterize all classes of structures where CSP(A, −) is solvable in polynomial time. Indeed, there are classes of structures having unbounded hypertree width that are tractable, because they have bounded fractional hypertree-width [52]. It seems that tighter results may be obtained by moving from polynomial-time tractability to fixed-parameter tractability (FPT).…”

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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A Join-Based Hybrid Parameter for Constraint Satisfaction

Ganian

Ordyniak

Szeider

2019

Lecture Notes in Computer Science

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We propose joinwidth, a new complexity parameter for the Constraint Satisfaction Problem (CSP). The definition of joinwidth is based on the arrangement of basic operations on relations (joins, projections, and pruning), which inherently reflects the steps required to solve the instance. We use joinwidth to obtain polynomial-time algorithms (if a corresponding decomposition is provided in the input) as well as fixed-parameter algorithms (if no such decomposition is provided) for solving the CSP. Joinwidth is a hybrid parameter, as it takes both the graphical structure as well as the constraint relations that appear in the instance into account. It has, therefore, the potential to capture larger classes of tractable instances than purely structural parameters like hypertree width and the more general fractional hypertree width (fhtw). Indeed, we show that any class of instances of bounded fhtw also has bounded joinwidth, and that there exist classes of instances of bounded joinwidth and unbounded fhtw, so bounded joinwidth properly generalizes bounded fhtw. We further show that bounded joinwidth also properly generalizes several other known hybrid restrictions, such as fhtw with degree constraints and functional dependencies. In this sense, bounded joinwidth can be seen as a unifying principle that explains the tractability of several seemingly unrelated classes of CSP instances.⋆ Robert Ganian acknowledges support by the Austrian Science Fund (FWF, Project P31336) and is also affiliated with FI MUNI, Czech Republic.

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Constraint Solving via Fractional Edge Covers

Cited by 138 publications

References 48 publications

Constructing NP-intermediate problems by blowing holes with parameters of various properties

Constructing NP-intermediate problems by blowing holes with parameters of various properties

Treewidth and Hypertree Width

A Join-Based Hybrid Parameter for Constraint Satisfaction

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