The notion of treewidth has seen to be a powerful vehicle for many graph algorithmic studies. This survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class. Also, some mutual relations between such classes are discussed.

This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, cross-references will be given to that book and the list of problems (NP-complete and harder) presented there. Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.) or open problems they would like publicized, should

. The problem instance of Vertex Cover consists of an undirected graph G = (V; E) and a positive integer k, the question is whether there exists a subset C V of vertices such that each edge in E has at least one of its endpoints in C with jCj k. We improve two recent worst case upper bounds for Vertex Cover. First, Balasubramanian et al. showed that Vertex Cover can be solved in time O(kn + 1:32472 k k 2 ), where n is the number of vertices in G. Afterwards, Downey et al. improved this to O(kn+ 1:31951 k k 2 ). Bringing the exponential base significantly below 1:3, we present the new upper bound O(kn + 1:29175 k k 2 ). 1 Introduction Vertex Cover is a problem of central importance in computer science: { It was among the rst NP-complete problems [7]. { There have been numerous eorts to design ecient approximation algorithms [3], but it is also known to be hard to approximate [1]. { It is of central importance in parameterized complexity theory and has one ...

Algorithmic methods based on the theory of fixed-parameter tractability are combined with powerful computational platforms to launch systematic attacks on combinatorial problems of significance. As a case study, optimal solutions to very large instances of the NP-hard vertex cover problem are computed. To accomplish this, an efficient sequential algorithm and various forms of parallel algorithms are devised, implemented and compared. The importance of maintaining a balanced decomposition of the search space is shown to be critical to achieving scalability. Target problems need only be amenable to reduction and decomposition. Applications in high throughput computational biology are also discussed.

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 present a linear time algorithm which, given a tree, computes a linear layout optimal with respect to vertex separation. As a consequence optimal edge search strategies, optimal node search strategies, and optimal interval augmentations can be computed also in O(n) for trees. This improves the running time of former algorithms from O(n log n) to O(n) and answers two related open questions raised in [7] and [15].

A novel combination of emergent algorithmic methods, powerful computational platforms and supporting infrastructure is described. These complementary tools and technologies are used to launch systematic attacks on combinatorial problems of significance. As a case study, optimal solutions to very large instances of the N P-hard vertex cover problem are computed. To accomplish this, an efficient sequential algorithm and two forms of parallel algorithms are implemented. The importance of maintaining a balanced decomposition of the search space is shown to be critical to achieving scalability. With the synergistic combination of techniques detailed here, it is now possible to solve problem instances that before were widely viewed as hopelessly out of reach. Target problems need only be amenable to reduction and decomposition. Applications are also discussed.

Abstract. Algorithms for series-parallel graphs can be extended to arbitrary two-terminal dags if node reductions are used along with series and parallel reductions. A node reduction contracts a vertex with unit in-degree (out-degree) into its sole incoming (outgoing) neighbor. This paper gives an O(n2"5) algorithm for minimizing node reductions, based on vertex cover in a transitive auxiliary graph. Applications include the analysis of PERT networks, dynamic programming approaches to network problems, and network reliability. For NP-hard problems one can obtain algorithms that are exponential only in the minimum number of node reductions rather than the number of vertices. This gives improvements if the underlying graph is nearly series-parallel.

By Robertson and Seymour’s graph minor theorem, every minor ideal can be characterised by a finite family of excluded minors. (A minor ideal is a class of graphs closed under taking minors.) We study algorithms for computing excluded minor characterisations of minor ideals. We propose a general method for obtaining such algorithms, which is based on definability in monadic second-order logic and the decidability of the monadic second-order theory of trees. A straightforward application of our method yields algorithms that, for a given k, compute excluded minor characterisations for the minor ideal Tk of all graphs of tree width at most k, the minor ideal Bk of all graphs of branch width at most k, and the minor ideal Gk of all graphs of genus at most k. Our main results are concerned with constructions of new minor ideals from given ones. Answering a question that goes back to Fellows and Langston [11], we prove that there is an algorithm that, given excluded minor characterisations of two minor ideals C and D, computes such a characterisation for the ideal C ∪ D. Furthermore, we obtain an algorithm for computing an excluded minor characterisation for the class of all apex graphs over a minor ideal C, given an excluded minor characterisation for C. (An apex graph over C is a graph G from which one vertex can be removed to obtain a graph in C.) A corollary of this result is a uniform ftpalgorithm for the “distance k from planarity” problem.

Recent time has seen quite some progress in the development of exponential time algorithms for NP-hard problems, where the base of the exponential term is fairly small. These developments are also tightly related to the theory of fixed parameter tractability. In this incomplete survey, we explain some basic techniques in the design of efficient fixed parameter algorithms, discuss deficiencies of parameterized complexity theory, and try to point out some future research challenges. The focus of this paper is on the design of efficient algorithms and not on a structural theory of parameterized complexity. Moreover, our emphasis will be laid on two exemplifying issues: Vertex Cover and MaxSat problems. A shorter version of this paper appears as an invited talk in the proceedings of the 25th Conference on Current Trends in Theory and Practice of Informatics (SOFSEM'98), Springer, LNCS , held in Jasna, Slovakia, November 21--27, 1998. y Supported by a Feodor Lynen fellowship of the Alex...