We present an exact algorithm that decides, for every fixed r ≥ 2 in time O(m) + 2 O(k 2 ) 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 Max-rSat is fixed-parameter 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 polynomial-time 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 above-mentioned parameterized Max-r-Sat admits a polynomial-size kernel.Combining another probabilistic argument with tools from graph matching theory and signed graphs, we show that if an instance of Max-2-Sat 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 fixed-parameter tractability and polynomial-size kernel results on Max-r-Sat can be extended to more general families of Boolean Constraint Satisfaction Problems. * A preliminary version of this paper is

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International audienceIn this paper, we show that the problems Disjoint Cycles and Disjoint Paths do not have polynomial kernels, unless N P ⊆ c o N P / p o l y . Thus, these problems do not allow polynomial time preprocessing that results in instances whose size is bounded by a polynomial in the parameter at hand. We build upon recent results by Bodlaender et al. [6] and Fortnow and Santhanam [20], that show that NP-complete problems that are ‘or-compositional’ do not have polynomial kernels, unless N P ⊆ c o N P / p o l y . To this machinery, we add a notion of transformation, and obtain that Disjoint Cycles, and Disjoint Paths do not have polynomial kernels, unless N P ⊆ c o N P / p o l y . For the proof, we introduce a problem on strings, called Disjoint Factors, and first show that this problem has no polynomial kernel unless N P ⊆ c o N P / p o l y . We also show that the related Disjoint Cycles Packing problem has a kernel of size O ( k log k )

A workflow specification defines a set of steps and the order in which these steps must be executed. Security requirements may impose constraints on which groups of users are permitted to perform subsets of these steps. A workflow specification is said to be satisfiable if there exists an assignment of users to workflow steps that satisfies all the constraints. An algorithm for determining whether such an assignment exists is important, both as a static analysis tool for workflow specifications and for the construction of runtime reference monitors for workflow management systems. Finding such an assignment is a hard problem in general, but work by Wang and Li [2010] using the theory of parameterized complexity suggests that efficient algorithms exist under reasonable assumptions about workflow specifications. In this article, we improve the complexity bounds for the workflow satisfiability problem. We also generalize and extend the types of constraints that may be defined in a workflow specification and prove that the satisfiability problem remains fixed-parameter tractable for such constraints. Finally, we consider preprocessing for the problem and prove that in an important special case, in polynomial time, we can reduce the given input into an equivalent one where the number of users is at most the number of steps. We also show that no such reduction exists for two natural extensions of this case, which bounds the number of users by a polynomial in the number of steps, provided a widely accepted complexity-theoretical assumption holds.

We present an exact algorithm that decides, for every fixed Our algorithm is based on a polynomial-time 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.Combining another probabilistic argument with tools from graph matching theory and signed graphs, we show that if an instance of Max-2-Sat 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 fixed-parameter tractability result on Max-r-Sat can be extended to a family of Boolean Constraint Satisfaction Problems.