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

EditorialThe traveling salesman problem This Special Issue of Discrete Optimization is devoted to the traveling salesman problem (TSP), its generalizations and modifications. The TSP is a very well studied optimization problem and there have been numerous publications on the TSP and its modifications, including two edited books (the latter was published in 2002 with G. Gutin and A.P. Punnen as editors). The topic continues to attract the attention of researchers from various areas including mathematics and computer science. Generally, TSP publications are scattered among various journals in mathematics, computer science, operations research, artificial intelligence and other related areas. The aim of this Special Issue is to bring to the attention of the reader a selected sample of research papers on the topic. We briefly describe below various papers included in this collection.It is well known that an asymmetric TSP on n vertices could be formulated as a symmetric TSP on 2n vertices. However, the paper by E. Balas, R. Carr, M. Fischetti and N. Simonetti included in this issue is the first article to demonstrate that the transformation can be used to obtain new facets for the symmetric TS polytope from asymmetric TS polytope facets. investigate a new class of polynomial length formulations for the asymmetric TSP. They show that a relaxation of their formulation is tighter than the formulation based on the exponential number of Danzig-Fulkerson-Johnson subtour elimination constraints. They also provide the results of computational experiments showing the efficiency of the proposed formulations.V. Mak and N. Boland study a recently defined generalization of the asymmetric TSP in which all arcs are partitioned into two classes (ordinary and replenishment arcs) and every vertex has a positive weight. A tour is feasible if it uses either type of arc before the total weight of vertices exceeds a certain limit, in which case only a replenishment arc can be used. The authors show that two classes of inequalities, under certain conditions, are facet-defining.A. Grigoriev and J. van de Klundert introduce another generalization of the TSP in which every vertex can be visited several times. The authors investigate how an optimal tour and its value change when the number of visits of each vertex is increased by the same factor.M. Turkensteen, D. Gosh, B. Goldengorin and G. Sierksma discuss an iterative technique for patching within the context of branch-and-bound approaches for the asymmetric TSP. Although already relevant per se, the technique could also be beneficial in other areas such as metaheuristics for the asymmetric TSP.Usually neighborhoods used in local search heuristics are of small polynomial size. This limitation is due to the fact that the best tour in the neighborhood has to be found for the worst case. However, as far back as the 1980s some researchers introduced TSP neighborhoods of exponential size in which the best tour can be computed in polynomial time. The study of exponential neighborhoods for various...

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The \emph{Workflow Satisfiability Problem (WSP)} is a problem of practical interest that arises whenever tasks need to be performed by authorized users, subject to constraints defined by business rules. We are required to decide whether there exists a \emph{plan} -- an assignment of tasks to authorized users -- such that all constraints are satisfied. Several bespoke algorithms have been constructed for solving the WSP, optimised to deal with constraints (business rules) of particular types. It is natural to see the WSP as a subclass of the {\em Constraint Satisfaction Problem (CSP)} in which the variables are tasks and the domain is the set of users. What makes the WSP distinctive as a CSP is that we can assume that the number of tasks is very small compared to the number of users. This is in sharp contrast with traditional CSP models where the domain is small and the number of variables is very large. As such, it is appropriate to ask for which constraint languages the WSP is fixed-parameter tractable (FPT), parameterized by the number of tasks. We have identified a new FPT constraint language, user-independent constraint, that includes many of the constraints of interest in business processing systems. We are also able to prove that the union of FPT languages remains FPT if they satisfy a simple compatibility condition. In this paper we present our generic algorithm, in which plans are grouped into equivalence classes, each class being associated with a \emph{pattern}. We demonstrate that our generic algorithm has running time $O^*(2^{k\log k})$, where $k$ is the number of tasks, for the language of user-independent constraints. We also show that there is no algorithm of running time $O^*(2^{o(k\log k)})$ for user-independent constraints unless the Exponential Time Hypothesis fails