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36
The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. 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 NPCompleteness,’ ’ W. H. Freeman & Co ..."
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Cited by 188 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. 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 NPCompleteness,’ ’ 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, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
Approximating the Bandwidth Via Volume Respecting Embeddings
, 1999
"... A linear arrangement of an nvertex graph is a onetoone 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 bandwidt ..."
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Cited by 93 (3 self)
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A linear arrangement of an nvertex graph is a onetoone 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 NPhard. 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 onetoone mapping f : V \Gamma! f1; 2; : : : ng, for which the bandwidth, i.e. max (i;j)2E jf(i) \Gamma f(j)j, i...
Four Strikes against Physical Mapping of DNA
 JOURNAL OF COMPUTATIONAL BIOLOGY
, 1993
"... 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 NPcomplete ..."
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Cited by 55 (8 self)
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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 NPcomplete decision problems: Colored unit interval graph completion, the maximum interval (or unit interval) subgraph, the pathwidth of a bipartite graph, and the kconsecutive 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.
Tractability of Parameterized Completion Problems on Chordal, Strongly Chordal and Proper Interval Graphs
, 1994
"... We study the parameterized complexity of three NPhard graph completion problems. The MINIMUM FILLIN 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 ..."
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Cited by 40 (5 self)
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We study the parameterized complexity of three NPhard graph completion problems. The MINIMUM FILLIN 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 bigO notation are small and do not depend on k. In particular, this implies that the problem is fixedparameter tractable (FPT). The PROPER
Pathwidth, Bandwidth and Completion Problems to Proper Interval Graphs with Small Cliques
 SIAM Journal on Computing
, 1996
"... 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 kcoloring c of G, does there exist a supergraph We show th ..."
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Cited by 29 (6 self)
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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 kcoloring 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 NPhard, 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.
SemiDefinite Relaxations for Minimum Bandwidth and other VertexOrdering problems
 THEOR. COMPUT. SCI
, 2000
"... We present simple semidefinite programming relaxations for the NPhard 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 vertexordering pr ..."
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Cited by 27 (4 self)
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We present simple semidefinite programming relaxations for the NPhard 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 vertexordering problems.
On Approximation Intractability of the Bandwidth Problem
, 1997
"... 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 p ..."
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Cited by 25 (0 self)
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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...
Polynomial Time Approximation Schemes for Some Dense Instances of NPHard Optimization Problems
 Proc. RANDOM 97, LNCS 1269
, 1997
"... We survey recent results on the existence of polynomial time approximation schemes for some dense instances of NPhard combinatorial optimization problems. We indicate further some inherent limits for existence of such schemes for some other dense instances of optimization problems. ..."
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Cited by 20 (8 self)
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We survey recent results on the existence of polynomial time approximation schemes for some dense instances of NPhard combinatorial optimization problems. We indicate further some inherent limits for existence of such schemes for some other dense instances of optimization problems.
Width parameters beyond treewidth and their applications
 Computer Journal
, 2007
"... Besides the very successful concept of treewidth (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth. These are special issues on Parameterized Complexity]), many concepts and parameters measuring the similarity or dissimilarity of structures compare ..."
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Cited by 19 (0 self)
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Besides the very successful concept of treewidth (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth. These are special issues on Parameterized Complexity]), many concepts and parameters measuring the similarity or dissimilarity of structures compared to trees have been born and studied over the past years. These concepts and parameters have proved to be useful tools in many applications, especially in the design of efficient algorithms. Our presented novel look at the contemporary developments of these ‘width ’ parameters in combinatorial structures delivers—besides traditional treewidth and derived dynamic programming schemes—also a number of other useful parameters like branchwidth, rankwidth (cliquewidth) or hypertreewidth. In this contribution, we demonstrate how ‘width ’ parameters of graphs and generalized structures (such as matroids or hypergraphs), can be used to improve the design of parameterized algorithms and the structural analysis in other applications on an abstract level.