Complexity and Approximability of Parameterized MAX-CSPs

Dell, Holger; Kim, Eun Jung; Lampis, Michael; Mitsou, Valia; Mömke, Tobias

doi:10.1007/s00453-017-0310-8

Cited by 7 publications

(11 citation statements)

References 41 publications

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“…A characterization of the kernelization complexity of min-ones CSPs parameterized by the number of variables was presented by Kratsch and Wahlström [26]. There are several parameterized complexity results for CSPs [8,10,25].…”

Section: Related Workmentioning

confidence: 99%

Optimal Sparsification for Some Binary CSPs Using Low-Degree Polynomials

Jansen

Pieterse

2019

ACM Trans. Comput. Theory

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This article analyzes to what extent it is possible to efficiently reduce the number of clauses in NP-hard satisfiability problems without changing the answer. Upper and lower bounds are established using the concept of kernelization. Existing results show that if NP ⊈ coNP/poly, no efficient preprocessing algorithm can reduce n -variable instances of cnf-sat with d literals per clause to equivalent instances with O ( n d −ε ) bits for any ε > 0. For the N ot -A ll -E qual sat problem, a compression to size Õ ( n d −1 ) exists. We put these results in a common framework by analyzing the compressibility of CSPs with a binary domain. We characterize constraint types based on the minimum degree of multivariate polynomials whose roots correspond to the satisfying assignments, obtaining (nearly) matching upper and lower bounds in several settings. Our lower bounds show that not just the number of constraints, but also the encoding size of individual constraints plays an important role. For example, for E xact S atisfiability with unbounded clause length it is possible to efficiently reduce the number of constraints to n +1, yet no polynomial-time algorithm can reduce to an equivalent instance with O ( n 2−ε ) bits for any ε > 0, unless NP ⊆ coNP/poly.

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Section: Related Workmentioning

confidence: 99%

Optimal Sparsification for Some Binary CSPs Using Low-Degree Polynomials

Jansen

Pieterse

2019

ACM Trans. Comput. Theory

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“…The paper [20] gave the first parameterized lower bounds on SAT with respect to graph width measures, in particular cliquewidth. This result was later improved to modular treewidth to complement an upper bound for model counting [21] and very recently to neighborhood diversity [11], a width measure introduced in [18]. We remark that the latter result could be turned into a conditional parameterized lower bound similar to that in [1] discussed above.…”

Section: Related Work Parameterized Knowledge Compilation Was First I...mentioning

confidence: 83%

Parameterized Compilation Lower Bounds for Restricted CNF-Formulas

Mengel

2016

Theory and Applications of Satisfiability Testing – SAT 2016

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We show unconditional parameterized lower bounds in the area of knowledge compilation, more specifically on the size of circuits in decomposable negation normal form (DNNF) that encode CNF-formulas restricted by several graph width measures. In particular, we show that• there are CNF formulas of size n and modular incidence treewidth k whose smallest DNNF-encoding has size n Ω(k) , and• there are CNF formulas of size n and incidence neighborhood diversity k whose smallest DNNF-encoding has size n Ω( √ k) .These results complement recent upper bounds for compiling CNF into DNNF and strengthen-quantitatively and qualitatively-known conditional lower bounds for cliquewidth. Moreover, they show that, unlike for many graph problems, the parameters considered here behave significantly differently from treewidth.

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“…The study of structural parameters which trade off the generality of treewidth for improved algorithmic properties is by now a standard topic in parameterized complexity. The most common type of work here is to consider a problem that is intractable parameterized by treewidth and see whether it becomes tractable parameterized by vertex cover or treedepth [2,10,13,16,17,31,32,35,34,36,42,41]. See [1] for a survey of results of this type.…”

Section: Related Workmentioning

confidence: 99%

Fine-grained Meta-Theorems for Vertex Integrity

Lampis¹,

Mitsou²

2021

Preprint

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Vertex Integrity is a graph measure which sits squarely between two more well-studied notions, namely vertex cover and tree-depth, and that has recently gained attention as a structural graph parameter. In this paper we investigate the algorithmic trade-offs involved with this parameter from the point of view of algorithmic meta-theorems for First-Order (FO) and Monadic Second Order (MSO) logic. Our positive results are the following: (i) given a graph G of vertex integrity k and an FO formula φ with q quantifiers, deciding if G satisfies φ can be done in time 2 O(k 2 q+q log q) + n O(1) ;(ii) for MSO formulas with q quantifiers, the same can be done in time 2 2 O(k 2 +kq)+ n O(1) . Both results are obtained using kernelization arguments, which pre-process the input to sizes 2 O(k 2 ) q and 2 O(k 2 +kq) respectively.The complexities of our meta-theorems are significantly better than the corresponding metatheorems for tree-depth, which involve towers of exponentials. However, they are worse than the roughly 2 O(kq) and 2 2 O (k+q) complexities known for corresponding meta-theorems for vertex cover. To explain this deterioration we present two formula constructions which lead to fine-grained complexity lower bounds and establish that the dependence of our meta-theorems on k is best possible. More precisely, we show that it is not possible to decide FO formulas with q quantifiers in time 2 o(k 2 q) , and that there exists a constant-size MSO formula which cannot be decided in time 2 2 o(k 2 ), both under the ETH. Hence, the quadratic blow-up in the dependence on k is unavoidable and vertex integrity has a complexity for FO and MSO logic which is truly intermediate between vertex cover and tree-depth.

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Complexity and Approximability of Parameterized MAX-CSPs

Cited by 7 publications

References 41 publications

Optimal Sparsification for Some Binary CSPs Using Low-Degree Polynomials

Optimal Sparsification for Some Binary CSPs Using Low-Degree Polynomials

Parameterized Compilation Lower Bounds for Restricted CNF-Formulas

Fine-grained Meta-Theorems for Vertex Integrity

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