Graph separation is a well-known tool to make (hard) graph problems accessible to a divide and conquer approach. We show how to use graph separator theorems in combination with (linear) problem kernels in order to develop xed parameter algorithms for many well-known NP-hard (planar) graph problems. We coin the key notion of glueable select&verify graph problems and derive from that a prospective way to easily check whether a planar graph problem will allow for a xed parameter algorithm of running time c p

The (unweighted) Maximum Satisfiability problem (MaxSat) is: given a boolean formula in conjunctive normal form, find a truth assignment that satisfies the most number of clauses. This paper describes exact algorithms that provide new upper bounds for MaxSat. We prove that MaxSat can be solved in time O(|F | 1.3803 K ), where |F | is the length of a formula F in conjunctive normal form and K is the number of clauses in F . We also prove the time bounds O(|F |1.3995 k ), where k is the maximum number of satisfiable clauses, and O(1.1279 |F | ) for the same problem. For Max2Sat this implies a bound of O(1.2722 K ). # An extended abstract of this paper was presented at the 26th International Colloquium on Automata, Languages, and Programming (ICALP'99), LNCS 1644, Springer-Verlag, pages 575--584, held in Prague, Czech Republic, July 11-15, 1999. + Supported by a Feodor Lynen fellowship (1998) of the Alexander von HumboldtStiftung, Bonn, and the Center for Discrete Ma...

In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A k-stack (resp...

The purposes of this paper are two: (1) to give an exposition of the main ideas of parameterized complexity, and (2) to discuss the connections of parameterized complexity to the systematic design of heuristics and approximation algorithms.

We describe a broad program of research in parameterized complexity, and hows this plays out for the MAX LEAF SPANNING TREE problem.

A set S of vertices in a digraph D = (V, A) is a kernel if S is independent and every vertex in V − S has an out-neighbor in S. We show that there exist O(n2 19.1 √ k + n 4)time and O(2 19.1 √ k k 9 + n 2)-time algorithms for checking whether a planar digraph D of order n has a kernel with at most k vertices. Moreover, if D has a kernel of size at most k, the algorithms find such a kernel of minimal size. 1

Fixed-parameter tractability(FPT) techniques have recently been successful in solving NP-complete problem instances of practical importance which were too large to be solved with previous methods. In this paper we show how to enhance this approach through the addition of parallelism, thereby allowing even larger problem instances to be solved in practice. More precisely, we demonstrate the potential of parallelism when applied to the bounded tree search phase of FPT algorithms. We apply our methodology to the k-Vertex Cover problem which has important applications, e.g., in multiple sequence alignments for computational biochemistry. We have implemented our parallel FPT method and application specific "plug-in" code for the k-Vertex Cover problem using C and the MPI communication library, and tested it on a network of 10 Sun SPARC workstations. This is the first experimental examination of parallel FPT techniques. In our experiments, we obtain excellent speedup results. Not only do we achieve a speedup of p in most cases, many cases even exhibit a super linear speedup. The latter result implies that our parallel methods, when simulated on a single processor, also yield a significant improvement over existing sequential methods.

Given a graph G = (V; E), Vertex Cover asks for a smallest subset V 0 ` V such that for each edge (a; b) in G a 2 V 0 or b 2 V 0 . We present an improved fixed-parameter tractable algorithm when the problem is parameterized by the size k of V 0 . The algorithm has a complexity of O(kn + maxf(1:25542) k k 2 ; (1:2906) k 2:5kg). We improve the klam value by 16 to k = 157. 1 Introduction In 1972, Karp has shown that the following problem is NP-complete [10]. Problem 1.1 Vertex Cover Instance: A graph G = (V; E), a positive integer k. Question: Does G have a vertex cover of size k? (I.e. does there exist a subset V 0 ` V , jV j k, such that for each (x; y) 2 E either x or y belongs to V 0 ?) Though NP-complete, the following parameterized version was one of the first problems shown to be fixed-parameter tractable [6, 9]. Problem 1.2 k-Vertex Cover Instance: A graph G = (V; E), a positive integer k. Parameter: k. Question: Does G have a vertex cover of size k?...

We consider the parameterized problem, whether for a given set D of n disks (of bounded radius ratio) in the Euclidean plane there exists a set of k non-intersecting disks. We expose an algorithm running in time n , that is|to our knowledge|the rst algorithm for this problem with running time bounded by an exponential with a sublinear exponent. For -precision disk graphs of bounded radius ratio, we show that the problem is xed parameter tractable with respect to parameter k.

In this paper we present improved exact and parameterized algorithms for the maximum satisfiability problem. In particular, we give an algorithm that computes a truth assignment for a boolean formula F satisfying the maximum number of clauses in time O(1.3247 m |F |), where m is the number of clauses in F, and |F | is the sum of the number of literals appearing in each clause in F. Moreover, given a parameter k, we give an O(1.3695 k + |F |) parameterized algorithm that decides whether a truth assignment for F satisfying at least k clauses exists. Both algorithms improve the previous best algorithms by Bansal and Raman for the problem. Key words. maximum satisfiability, exact algorithms, parameterized algorithms. 1