Parameterizing MAX SNP Problems Above Guaranteed Values

Mahajan, Meena; Raman, Venkatesh; Sikdar, Somnath

doi:10.1007/11847250_4

Cited by 7 publications

(5 citation statements)

References 12 publications

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“…Indeed, W/2 is the average weight of satisfied equations (as the probability of each equation to be satisfied is 1/2) and, thus, is a lower bound; to see the tightness consider a system of pairs of equations of the form i∈I z i = 0, i∈I z i = 1 of weight 1. Mahajan et al [13,14] parameterized Max Lin as follows: given a Max Lin system Az = b, decide whether the total weight of satisfied equations minus W/2 is at least k ′ , where W is the total weight of all equations and k ′ is the parameter. This is equivalent to asking whether the maximum excess is at least k, where k = 2k ′ is the parameter.…”

Section: Introductionmentioning

confidence: 99%

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Systems of Linear Equations over $\mathbb{F}_2$ and Problems Parameterized above Average

Crowston

Gutin

Jones

et al. 2010

Lecture Notes in Computer Science

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In the problem Max Lin, we are given a system Az = b of m linear equations with n variables over F2 in which each equation is assigned a positive weight and we wish to find an assignment of values to the variables that maximizes the excess, which is the total weight of satisfied equations minus the total weight of falsified equations. Using an algebraic approach, we obtain a lower bound for the maximum excess.Max Lin Above Average (Max Lin AA) is a parameterized version of Max Lin introduced by Mahajan et al. (Proc. IWPEC'06 and J. Comput. Syst. Sci. 75, 2009). In Max Lin AA all weights are integral and we are to decide whether the maximum excess is at least k, where k is the parameter.It is not hard to see that we may assume that no two equations in Az = b have the same left-hand side and n = rankA. Using our maximum excess results, we prove that, under these assumptions, Max Lin AA is fixed-parameter tractable for a wide special case: m ≤ 2 p(n) for an arbitrary fixed function p(n) = o(n). This result generalizes earlier results by Crowston et al. (arXiv:0911.5384) and Gutin et al. (Proc. IWPEC'09). We also prove that Max Lin AA is polynomial-time solvable for every fixed k and, moreover, Max Lin AA is in the parameterized complexity class W [P].Max r-Lin AA is a special case of Max Lin AA, where each equation has at most r variables. In Max Exact r-SAT AA we are given a multiset of m clauses on n variables such that each clause has r variables and asked whether there is a truth assignment to the n variables that satisfies at least (1 − 2 −r )m + k2 −r clauses. Using our maximum excess results, we prove that for each fixed r ≥ 2, Max r-Lin AA and Max Exact r-SAT AA can be solved in time 2 It is easy to see that maximization of arbitrary pseudo-boolean functions, i.e., functions f : {−1, +1} n → R, represented by their Fourier expansions is equivalent to solving Max Lin. Using our main maximum excess result, we obtain a tight lower bound on the maxima of pseudo-boolean functions.

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Section: Introductionmentioning

confidence: 99%

“…Mahajan et al [13,14] raised the question of determining the parameterized complexity of Max Lin AA. It is not hard to see (we explain it in detail in Section 2) that we may assume that no two equations in Az = b have the same left-hand side and n = rankA.…”

Section: Introductionmentioning

confidence: 99%

Systems of Linear Equations over $\mathbb{F}_2$ and Problems Parameterized above Average

Crowston

Gutin

Jones

et al. 2010

Lecture Notes in Computer Science

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“…In terms of parameterized complexity, since a minimum vertex cover of an n-vertex graph with perfect matching contains at least n/2 vertices, it seems more reasonable to parameterize the problem by studying whether such a graph has a vertex cover of size bounded by n/2+k, where k is the parameter [18][19][20]. This way of problem parameterization has been named "parameterizing above or below guaranteed values" in [20], which has received increasing attention recently.…”

Section: Introductionmentioning

confidence: 99%

On the parameterized vertex cover problem for graphs with perfect matching

Wang

et al. 2014

Sci. China Inf. Sci.

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“…For more background on kernelization we refer to the recent surveys on kernelization given by Guo and Niedermeier [23] as well as by Bodlaender [4]. In an earlier paper, Mahajan et al [31] studied MAX SNP problems and observe that kernelizations follow from the fact that NP-hard problems in MAX SNP have guaranteed lower bounds for the optimum value, motivating them to study these problems parameterized above such lower bounds. Cai and Huang [9] showed that all problems in MAX SNP admit fixed-parameter approximation schemes.…”

Section: Introductionmentioning

confidence: 99%

Polynomial Kernelizations for MIN F+Π1 and MAX NP

Kratsch

2011

Algorithmica

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It has been observed in many places that constant-factor approximable problems often admit polynomial or even linear problem kernels for their decision versions, e.g., VERTEX COVER, FEEDBACK VERTEX SET, and TRIANGLE PACK-ING. While there exist examples like BIN PACKING, which does not admit any kernel unless P = NP, there apparently is a strong relation between these two polynomialtime techniques. We add to this picture by showing that the natural decision versions of all problems in two prominent classes of constant-factor approximable problems, namely MIN F + 1 and MAX NP, admit polynomial problem kernels. Problems in MAX SNP, a subclass of MAX NP, are shown to admit kernels with a linear base set, e.g., the set of vertices of a graph. This extends results of Cai and Chen (J. Comput. Syst. Sci. 54(3): 465-474, 1997), stating that the standard parameterizations of problems in MAX SNP and MIN F + 1 are fixed-parameter tractable, and complements recent research on problems that do not admit polynomial kernelizations (Bodlaender et al. in J. Comput. Syst. Sci. 75(8): 423-434, 2009).

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Parameterizing MAX SNP Problems Above Guaranteed Values

Cited by 7 publications

References 12 publications

Systems of Linear Equations over $\mathbb{F}_2$ and Problems Parameterized above Average

Systems of Linear Equations over $\mathbb{F}_2$ and Problems Parameterized above Average

On the parameterized vertex cover problem for graphs with perfect matching

Polynomial Kernelizations for MIN F+Π1 and MAX NP

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