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Parameterizing MAX SNP Problems Above Guaranteed Values
 Proc. of IWPEC, Springer LNCS 4169
, 2006
"... Abstract. We show that every problem in MAX SNP has a lower bound on the optimum solution size and that the above guarantee question with respect to that lower bound is fixed parameter tractable. We next introduce the notion of ‘tight ’ upper and lower bounds for the optimum solution and show that t ..."
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Abstract. We show that every problem in MAX SNP has a lower bound on the optimum solution size and that the above guarantee question with respect to that lower bound is fixed parameter tractable. We next introduce the notion of ‘tight ’ upper and lower bounds for the optimum solution and show that the parameterized version of a variant of the above guarantee question with respect to the tight lower bound cannot be fixed parameter tractable unless P = NP, for a number of NPoptimization problems. 1
Abstract Maximizing quadratic programs: extending Grothendieck’s inequality
"... This paper considers the following type of quadratic programming problem. Given an arbitrary matrix A, whose diagonal elements are zero, find x ∈ {−1, 1} n such that x T Ax is maximized. Our approximation algorithm for this problem uses the canonical semidefinite relaxation and returns a solution wh ..."
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This paper considers the following type of quadratic programming problem. Given an arbitrary matrix A, whose diagonal elements are zero, find x ∈ {−1, 1} n such that x T Ax is maximized. Our approximation algorithm for this problem uses the canonical semidefinite relaxation and returns a solution whose ratio to the optimum is in Ω(1 / logn). This quadratic programming problem can be seen as an extension to that of maximizing x T Ay (where y’s components are also ±1). Grothendieck’s inequality states that the ratio of the optimum value of the latter problem to the optimum of its canonical semidefinite relaxation is bounded below by a constant. The study of this type of quadratic program arose from a desire to approximate the maximum correlation in correlation clustering. Nothing substantive was known about this problem; we present an Ω(1 / logn) approximation, based on our quadratic programming algorithm. We can also guarantee that our quadratic programming algorithm returns a solution to the MAXCUT problem that has a significant advantage over a random assignment. 1.