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Solving MAXrSAT above a Tight Lower Bound
, 2010
"... We present an exact algorithm that decides, for every fixed r ≥ 2 in time O(m) + 2 O(k2) whether a given multiset of m clauses of size r admits a truth assignment that satisfies at least ((2 r − 1)m + k)/2 r clauses. Thus MaxrSat is fixedparameter tractable when parameterized by the number of sat ..."
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Cited by 21 (8 self)
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We present an exact algorithm that decides, for every fixed r ≥ 2 in time O(m) + 2 O(k2) whether a given multiset of m clauses of size r admits a truth assignment that satisfies at least ((2 r − 1)m + k)/2 r clauses. Thus MaxrSat is fixedparameter tractable when parameterized by the number of satisfied clauses above the tight lower bound (1 − 2 −r)m. This solves an open problem of Mahajan, Raman and Sikdar (J. Comput. System Sci., 75, 2009). Our algorithm is based on a polynomialtime data reduction procedure that reduces a problem instance to an equivalent algebraically represented problem with O(k 2) variables. This is done by representing the instance as an appropriate polynomial, and by applying a probabilistic argument combined with some simple tools from Harmonic analysis to show that if the polynomial cannot be reduced to one of size O(k 2), then there is a truth assignment satisfying the required number of clauses. We introduce a new notion of bikernelization from a parameterized problem to another one and apply it to prove that the abovementioned parameterized MaxrSat admits a polynomialsize kernel. Combining another probabilistic argument with tools from graph matching theory and signed graphs, we show that if an instance of Max2Sat with m clauses has at least 3k variables after application of certain polynomial time reduction rules to it, then there is a truth assignment that satisfies at least (3m + k)/4 clauses. We also outline how the fixedparameter tractability and polynomialsize kernel results on MaxrSat can be extended to more general families of Boolean
Parameterizing above or below guaranteed values
 J. Comput. System Sci
"... We consider new parameterizations of NPoptimization problems that have nontrivial lower and/or upper bounds on their optimum solution size. The natural parameter, we argue, is the quantity above the lower bound or below the upper bound. We show that for every problem in MAX SNP, the optimum value i ..."
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Cited by 20 (2 self)
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We consider new parameterizations of NPoptimization problems that have nontrivial lower and/or upper bounds on their optimum solution size. The natural parameter, we argue, is the quantity above the lower bound or below the upper bound. We show that for every problem in MAX SNP, the optimum value is bounded below by an unbounded function of the inputsize, and that the aboveguarantee parameterization with respect to this lower bound is fixedparameter tractable. We also observe that approximation algorithms give nontrivial lower or upper bounds on the solution size and that the above or below guarantee question with respect to these bounds is fixedparameter tractable for a subclass of NPoptimization problems. We then introduce the notion of ‘tight ’ lower and upper bounds and exhibit a number of problems for which the aboveguarantee and belowguarantee parameterizations with respect to a tight bound is fixedparameter tractable or Whard. We show that if we parameterize “sufficiently ” above or below the tight bounds, then these parameterized versions are not fixedparameter tractable unless P = NP, for a subclass of NPoptimization problems. We also list several directions to explore in this paradigm. 1
Algorithmic and complexity results for decompositions of biological networks into monotone subsystems
 IN LECTURE NOTES IN COMPUTER SCIENCE: EXPERIMENTAL ALGORITHMS: 5TH INTERNATIONAL WORKSHOP, WEA 2006, SPRINGERVERLAG, 253–264. (CALA GALDANA, MENORCA
, 2006
"... A useful approach to the mathematical analysis of largescale biological networks is based upon their decompositions into monotone dynamical systems. This paper deals with two computational problems associated to finding decompositions which are optimal in an appropriate sense. In graphtheoretic la ..."
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Cited by 14 (5 self)
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A useful approach to the mathematical analysis of largescale biological networks is based upon their decompositions into monotone dynamical systems. This paper deals with two computational problems associated to finding decompositions which are optimal in an appropriate sense. In graphtheoretic language, the problems can be recast in terms of maximal signconsistent subgraphs. The theoretical results include polynomialtime approximation algorithms as well as constantratio inapproximability results. One of the algorithms, which has a worstcase guarantee of 87.9 % from optimality, is based on the semidefinite programming relaxation approach of GoemansWilliamson [23]. The algorithm was implemented and tested on a Drosophila segmentation network and an Epidermal Growth Factor Receptor pathway model, and it was found to perform close to optimally.
A probabilistic approach to problems parameterized above tight lower bound. CoRR technical report
 University of Copenhagen
, 2009
"... We introduce a new approach for establishing fixedparameter tractability of problems parameterized above tight lower bounds or below tight upper bounds. To illustrate the approach we consider three problems of this type of unknown complexity that were introduced by Mahajan, Raman and Sikdar (J. Com ..."
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Cited by 12 (11 self)
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We introduce a new approach for establishing fixedparameter tractability of problems parameterized above tight lower bounds or below tight upper bounds. To illustrate the approach we consider three problems of this type of unknown complexity that were introduced by Mahajan, Raman and Sikdar (J. Comput. Syst. Sci. 75, 2009). We show that a generalization of one of the problems and nontrivial special cases of the other two are fixedparameter tractable. 1
Simultaneously Satisfying Linear Equations Over F2
 MaxLin2 and MaxrLin2 Parameterized Above Average. In FSTTCS 2011, LIPICS
"... ABSTRACT. In the parameterized problem MAXLIN2AA[k], we are given a system with variables x1,..., xn consisting of equations of the form ∏i∈I x i = b, where x i, b ∈ {−1, 1} and I ⊆ [n], each equation has a positive integral weight, and we are to decide whether it is possible to simultaneously sati ..."
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Cited by 3 (1 self)
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ABSTRACT. In the parameterized problem MAXLIN2AA[k], we are given a system with variables x1,..., xn consisting of equations of the form ∏i∈I x i = b, where x i, b ∈ {−1, 1} and I ⊆ [n], each equation has a positive integral weight, and we are to decide whether it is possible to simultaneously satisfy equations of total weight at least W/2 + k, where W is the total weight of all equations and k is the parameter (if k = 0, the possibility is assured). We show that MAXLIN2AA[k] has a kernel with at most O(k 2 log k) variables and can be solved in time 2 O(k log k) (nm) O(1). This solves an open problem of Mahajan et al. (2006). The problem MAXrLIN2AA[k, r] is the same as MAXLIN2AA[k] with two differences: each equation has at most r variables and r is the second parameter. We prove a theorem on MAXrLIN2AA[k, r] which implies that MAXrLIN2AA[k, r] has a kernel with at most (2k − 1)r variables, improving a number of results including one by Kim and Williams (2010). The theorem also implies a lower bound on the maximum of a function f: {−1, 1} n → R whose Fourier expansion (which is a multilinear polynomial) is of degree r. We show applicability of the lower bound by giving a new proof of the EdwardsErdős bound (each connected graph on n vertices and m edges has a bipartite subgraph with at least m/2 + (n − 1)/4 edges) and obtaining a generalization. 1
The Approximability of Learning and Constraint Satisfaction Problems
, 2010
"... International Business Machine. The views and conclusions contained in this document are those of the ..."
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International Business Machine. The views and conclusions contained in this document are those of the