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

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A linear arrangement of an n-vertex graph is a one-to-one mapping of its vertices to the integers f1; : : : ; ng. The bandwidth of a linear arrangement is the maximum difference between mapped values of adjacent vertices. The problem of finding a linear arrangement with smallest possible bandwidth in NP-hard. We present a randomized algorithm that runs in nearly linear time and outputs a linear arrangement whose bandwidth is within a polylogarithmic multiplicative factor of optimal. Our algorithm is based on a new notion, called volume respecting embeddings, which is a natural extension of small distortion embeddings of Bourgain and of Linial, London and Rabinovich. 1 Introduction We consider the problem of minimizing the bandwidth of an undirected connected graph G(V; E), where n = jV j and m = jEj. One needs to find a linear arrangement of the vertices, namely, a one-to-one mapping f : V \Gamma! f1; 2; : : : ng, for which the bandwidth, i.e. max (i;j)2E jf(i) \Gamma f(j)j, i...

Physical Mapping is a central problem in molecular biology ... and the human genome project. The problem is to reconstruct the relative position of fragments of DNA along the genome from information on their pairwise overlaps. We show that four simplified models of the problem lead to NP-complete decision problems: Colored unit interval graph completion, the maximum interval (or unit interval) subgraph, the pathwidth of a bipartite graph, and the k-consecutive ones problem for k >= 2. These models have been chosen to reflect various features typical in biological data, including false negative and positive errors, small width of the map and chimericism.

We study the parameterized complexity of three NP-hard graph completion problems. The MINIMUM FILL-IN problem is to decide if a graph can be triangulated by adding at most k edges. We develop O(c m) and O(k mn + f(k)) algorithms for this problem on a graph with n vertices and m edges. Here f(k) is exponential in k and the constants hidden by the big-O notation are small and do not depend on k. In particular, this implies that the problem is fixed-parameter tractable (FPT). The PROPER

We study two related problems motivated by molecular biology: ffl Given a graph G and a constant k, does there exist a supergraph G of G which is a unit interval graph and has clique size at most k? ffl Given a graph G and a proper k-coloring c of G, does there exist a supergraph We show that those problems are polynomial for fixed k. On the other hand we prove that the first problem is equivalent to deciding if the bandwidth of G is at most k \Gamma 1. Hence, it is NP-hard, and W [t]-hard for all t. We also show that the second problem is W [1]-hard. This implies that for fixed k, both of the problems are unlikely to have an O(n ) algorithm, where ff is a constant independent of k.

We present simple semi-definite programming relaxations for the NP-hard minimum bandwidth and minimum length linear ordering problems. We then show how these relaxations can be rounded in a natural way (via random projection) to obtain approximation guarantees for both of these vertex-ordering problems.

The bandwidth problem is the problem of enumerating the vertices of a given graph G such that the maximum difference between the numbers of adjacent vertices is minimal. The problem has a long history and a number of applications. There was not much known though on approximation hardness of this problem, till recently. Karpinski and Wirtgen [KW 97] showed that there are no polynomial time approximation algorithms with an absolute error guarantee of n 1\Gammaffl for any ffl ? 0 unless P = NP . In this paper we show, that there is no PTAS for the bandwidth problem unless P = NP , even for trees. More precisely we show that there are no polynomial time approximation algorithms for general graphs with an approximation ratio better than 1:5, and for the trees with an approximation ratio better than 4=3 ß 1:332. Dept. of Computer Science, University of Bonn, 53117 Bonn. Email: blache@cs.bonn.edu. y Dept. of Computer Science, University of Bonn, 53117 Bonn, and International Compute...

We survey recent results on the existence of polynomial time approximation schemes for some dense instances of NP-hard combinatorial optimization problems. We indicate further some inherent limits for existence of such schemes for some other dense instances of optimization problems.

This chapter gives a general overview of two emerging techniques for discrete optimization that have footholds in mathematics, computer science, and operations research: branch decompositions and tree decompositions. Branch decompositions and tree decompositions along with their respective connectivity invariants, branchwidth and treewidth, were first introduced to aid in proving the Graph Minors Theorem, a wellknown conjecture (Wagner’s conjecture) in graph theory. The algorithmic importance of branch decompositions and tree decompositions for solving NP-hard problems modelled on graphs was first realized by computer scientists in relation to formulating graph problems in monadic second order logic. The dynamic programming techniques utilizing branch decompositions and tree decompositions, called branch decomposition and tree decomposition based algorithms, fall into a class of algorithms known as fixed-parameter tractable algorithms and have been shown to be effective in a practical setting for NP-hard problems such as minimum domination, the travelling salesman problem, general minor containment, and frequency assignment problems.