Parameterized Complexity and Kernelizability of Max Ones and Exact Ones Problems

Kratsch, Stefan; Marx, Dániel; Wahlström, Magnus

doi:10.1007/978-3-642-15155-2_43

Cited by 22 publications

(20 citation statements)

References 17 publications

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“…There have been further parameterized complexity studies of Boolean CSP [21,29,22,20], but CSP's with larger domains (i.e., where the variables are not Boolean) were not studied. In most cases, we expect that results for larger domains are much more complex than for the Boolean case, and usually require significant new ideas (compare e.g., Schaefer's Theorem [28] with the 3-element version [5]).…”

Section: Introductionmentioning

confidence: 99%

Constraint Satisfaction Parameterized by Solution Size

Bulatov¹,

Marx²

2014

SIAM J. Comput.

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In the constraint satisfaction problem (CSP) corresponding to a constraint language (i.e., a set of relations) Γ, the goal is to find an assignment of values to variables so that a given set of constraints specified by relations from Γ is satisfied. The complexity of this problem has received substantial amount of attention in the past decade. In this paper, we study the fixed-parameter tractability of constraint satisfaction problems parameterized by the size of the solution in the following sense: one of the possible values, say 0, is "free," and the number of variables allowed to take other, "expensive," values is restricted. A size constraint requires that exactly k variables take nonzero values. We also study a more refined version of this restriction: a global cardinality constraint prescribes how many variables have to be assigned each particular value. We study the parameterized complexity of these types of CSPs where the parameter is the required number k of nonzero variables. As special cases, we can obtain natural and well-studied parameterized problems such as Independent set, Vertex Cover, d-Hitting Set, Biclique, etc.In the case of constraint languages closed under substitution of constants, we give a complete characterization of the fixed-parameter tractable cases of CSPs with size constraints, and we show that all the remaining problems are W[1]-hard. For CSPs with cardinality constraints, we obtain a similar classification, but for some of the problems we are only able to show that they are Biclique-hard. The exact parameterized complexity of the Biclique problem is a notorious open problem, although it is believed to be W[1]-hard.

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Section: Introductionmentioning

confidence: 99%

Constraint Satisfaction Parameterized by Solution Size

Bulatov¹,

Marx²

2014

SIAM J. Comput.

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“…Using their schema, Dom et al [11] were able to show that important problems such as Connected Vertex Cover and Subset Sum are unlikely to have polynomial kernels. Later their technique was used for showing several important results, including dichotomy theorems for CSP kernelization [22,24].…”

Section: Introductionmentioning

confidence: 99%

Weak Compositions and Their Applications to Polynomial Lower Bounds for Kernelization

Hermelin¹,

Wu²

2012

Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms

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In this paper we use the notion of weak compositions to obtain polynomial kernelization lower-bounds for several natural parameterized problems. Let d ≥ 2 be some constant and let L 1 , L 2 ⊆ {0, 1} * × N be two parameterized problems where the unparameterized version of L 1 is NP-hard. Assuming coNP ⊆ NP/poly, our framework essentially states that composing t L 1 -instances each with parameter k, to an L 2 -instance with parame-We show two examples of weak composition and derive polynomial kernelization lower bounds for d-Bipartite Regular Perfect Code and d-Dimensional Matching, parameterized by the solution size k. By reduction, using linear parameter transformations, we then derive the following lower-bounds for kernel sizes when the parameter is the solution size k (assuming coNP ⊆ NP/poly):Cover, Hitting Set with d-Bounded Occurrences, and Exact Hitting Set with dBounded Occurrences have no kernels of size O(k d−3−ε ) for any ε > 0.• K d Packing and Induced K 1,d Packing have no kernels of size O(k d−4−ε ) for any ε > 0.• Our results give a negative answer to an open question raised by Dom, Lokshtanov, and * The full version of this paper can be found at [21].† Contact: Max Plank Institute for Informatics, Germany, Email: hermelin@mpi-inf.mpg.de.‡ Contact: University of Wisconsin-Madison, USA, Email: xiwu@cs.wisc.edu.Work done during the research visit at Max Planck Institute for Informatics, Germany. Research partially supported by NSF grant 1017597, NSFC grant 60973026, the Shanghai Leading Academic Discipline Project (no. B114), and the Shanghai Committee of Science and Technology (nos. 08DZ2271800 and 09DZ2272800).Saurabh [ICALP2009] regarding the existence of uniform polynomial kernels for the problems above. All our lower bounds transfer automatically to compression lower bounds, a notion defined by Harnik and Naor [SICOMP2010] to study the compressibility of NP instances with cryptographic applications. We believe weak composition can be used to obtain polynomial kernelization lower bounds for other interesting parameterized problems.In the last part of the paper we strengthen previously known super-polynomial kernelization lower bounds to super-quasi-polynomial lower bounds, by showing that quasi-polynomial kernels for compositional NP-hard parameterized problems implies the collapse of the exponential hierarchy. These bounds hold even the kernelization algorithms are allowed to run in quasipolynomial time.

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“…By abuse of notation, let us denote the corresponding parameterized weighted satisfiability problems by p-WSAT = (S), p-WSAT ≤ (S), p-WSAT ≥ (S) (using S instead of B thus signals that we consider constraint satisfaction problems). It is known that p-WSAT ≤ (S) is in FPT for all S [10]; the complexity of p-WSAT = (S) is given by a dichotomy that does not obey the Schaefer classes or even more general co-clone classes [12], and the complexity of p-WSAT ≥ (S) ranges from polynomial-time to para-NP-completeness. When we study these decision problems from a classical non-parameterized complexity point of view, the complexity differs, cf.…”

Section: Resultsmentioning

confidence: 99%

Parameterized Complexity of Weighted Satisfiability Problems: Decision, Enumeration, Counting

Creignou

Vollmer

2015

Fundamenta Informaticae

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We consider the weighted satisfiability problem for Boolean circuits and propositional formulae, where the weight of an assignment is the number of variables set to true. We study the parameterized complexity of these problems and initiate a systematic study of the complexity of its fragments. Only the monotone fragment has been considered so far and proven to be of same complexity as the unrestricted problems. Here, we consider all fragments obtained by semantically restricting circuits or formulae to contain only gates (connectives) from a fixed set B of Boolean functions. We obtain a dichotomy result by showing that for each such B, the weighted satisfiability problems are either W[P]-complete (for circuits) or W[SAT]-complete (for formulae) or efficiently solvable. We also consider the related enumeration and counting problems.

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Parameterized Complexity and Kernelizability of Max Ones and Exact Ones Problems

Cited by 22 publications

References 17 publications

Constraint Satisfaction Parameterized by Solution Size

Constraint Satisfaction Parameterized by Solution Size

Weak Compositions and Their Applications to Polynomial Lower Bounds for Kernelization

Parameterized Complexity of Weighted Satisfiability Problems: Decision, Enumeration, Counting

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