The maximum satisfiability problem (MAX-SAT) is stated as follows: Given Boolean formula in CNF, find a truth assignment that satisfies the maximum possible number of its clauses. MAX-SAT is MAX-SNP-complete and received much attention recently. One of the challenges posed by Alber, Gramm and Niedermeier in a recent survey paper asks: Can MAX-SAT be solved in less than 2 ' "steps"? Here, n is the number of different variables in the formula and a step may take polynomial time of the input. We answered this challenge positively by showing that popular algorithm based on branch-and-bound is bounded by O(b2 ') in time, where b is the maximum number of occurrences of any variable in the input.

Our goal in this paper is the development of fast algorithms for recognizing general classes of graphs. We seek algorithms whose complexity can be expressed as a linear function of the graph size plus an exponential function of k, a natural parameter describing the class. In particular, we consider the class W_k(G), where for each graph G in W_k(G), the removal of a set of at most k vertices from G results in a graph in the base graph class G. (If G ist the class of edgeless graphs,...

The decision version of the maximum satisfiability problem (MAX-SAT) is stated as follows: Given a set S of propositional clauses and an integer g, decide if there exists a truth assignment that falsifies at most g clauses in S, where g is called the allowance for false clauses. We conduct an extensive experiment on over a million of random instances of 2-SAT and identify statistically the relationship between g, n (number of variables) and m (number of clauses). In our experiment, we apply an e#cient decision procedure based on the branch-and-bound method. The statistical data of the experiment confirm not only the "scaling window" of MAX-2-SAT discovered by Chayes, Kim and Borgs, but also the recent results of Coppersmith et al. While there is no easy-hard-easy pattern for the complexity of 2-SAT at the phase transition, we show that there is such a pattern for the decision problem of MAX-2-SAT associated with the phase transition. We also identify that the hardest problems are among those with high allowance for false clauses but low number of clauses.

Adequate choice of data structures and special effort in implementation are crucial to the good performance of parallel algorithms. In this paper, we present experimental results of a BSP/CGM implementation for the FPT (Fixed-Parameter Tractability) Vertex Cover problem, also known as k-Vertex Cover. We propose an alternative implementation that has as its basis an algorithm that combines the parallel FPT algorithm proposed by Cheetham et al. and the Downey’s et al. sequential FPT algorithm. Previously, a better and refined implementation, based on the Cheetham et al. Algorithm was presented by Hanashiro. In his experiments, Hanashiro obtained better results than those presented by Cheetham et al. In this paper, implemented the new adapted algorithm for the k-Vertex Cover and compared our experimental results with those of Hanashiro et al, using the same input data (conflict graphs of amino acids). We report substantial improvement over the results of Hanahiro et al, with speedups from 3 to 20 times relative to that implementation. 1.