Results 1  10
of
44
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 ..."
Abstract

Cited by 188 (0 self)
 Add to MetaCart
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
The Vertex Separation And Search Number Of A Graph
"... We relate two concepts in graph theory and algorithmic complexity, namely the search number and the vertex separation of a graph. Let s (G ) denote the search number and vs (G ) denote the vertex separation of a connected, undirected graph G . We show that vs (G ) s (G ) vs (G ) + 2 and we give a ..."
Abstract

Cited by 74 (1 self)
 Add to MetaCart
We relate two concepts in graph theory and algorithmic complexity, namely the search number and the vertex separation of a graph. Let s (G ) denote the search number and vs (G ) denote the vertex separation of a connected, undirected graph G . We show that vs (G ) s (G ) vs (G ) + 2 and we give a simple transformation from G to G such that vs (G ) = s (G ). We characterize those trees having a given vertex separation and describe the smallest such trees. We also note that there exist trees for which the difference between search number and vertex separation is indeed 2. We give algorithms that, for any tree T , compute vs (T ) in linear time and compute an optimal layout with respect to vertex separation in time O (n log n ). Vertex separation has previously been related to progressive black/white pebble demand and has been shown to be identical to a variant of search number, node search number, and to path width, which has been related directly to gate matrix layout cost. All these...
Parameterized Complexity: A Framework for Systematically Confronting Computational Intractability
 DIMACS Series in Discrete Mathematics and Theoretical Computer Science
, 1997
"... In this paper we give a programmatic overview of parameterized computational complexity in the broad context of the problem of coping with computational intractability. We give some examples of how fixedparameter tractability techniques can deliver practical algorithms in two different ways: (1) by ..."
Abstract

Cited by 67 (15 self)
 Add to MetaCart
In this paper we give a programmatic overview of parameterized computational complexity in the broad context of the problem of coping with computational intractability. We give some examples of how fixedparameter tractability techniques can deliver practical algorithms in two different ways: (1) by providing useful exact algorithms for small parameter ranges, and (2) by providing guidance in the design of heuristic algorithms. In particular, we describe an improved FPT kernelization algorithm for Vertex Cover, a practical FPT algorithm for the Maximum Agreement Subtree (MAST) problem parameterized by the number of species to be deleted, and new general heuristics for these problems based on FPT techniques. In the course of making this overview, we also investigate some structural and hardness issues. We prove that an important naturally parameterized problem in artificial intelligence, STRIPS Planning (where the parameter is the size of the plan) is complete for W [1]. As a corollary, this implies that kStep Reachability for Petri Nets is complete for W [1]. We describe how the concept of treewidth can be applied to STRIPS Planning and other problems of logic to obtain FPT results. We describe a surprising structural result concerning the top end of the parameterized complexity hierarchy: the naturally parameterized Graph kColoring problem cannot be resolved with respect to XP either by showing membership in XP, or by showing hardness for XP without settling the P = NP question one way or the other.
Parameterized Computational Feasibility
 Feasible Mathematics II
, 1994
"... Many natural computational problems have input consisting of two or more parts. For example, the input might consist of a graph and a positive integer. For many natural problems we may view one of the inputs as a parameter and study how the complexity of the problem varies if the parameter is he ..."
Abstract

Cited by 59 (20 self)
 Add to MetaCart
Many natural computational problems have input consisting of two or more parts. For example, the input might consist of a graph and a positive integer. For many natural problems we may view one of the inputs as a parameter and study how the complexity of the problem varies if the parameter is held fixed. For many applications of computational problems involving such a parameter, only a small range of parameter values is of practical significance, so that fixedparameter complexity is a natural concern. In studying the complexity of such problems, it is therefore important to have a framework in which we can make qualitative distinctions about the contribution of the parameter to the complexity of the problem. In this paper we survey one such framework for investigating parameterized computational complexity and present a number of new results for this theory.
Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring
 In 46th Annual IEEE Symposium on Foundations of Computer Science
, 2005
"... At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topolog ..."
Abstract

Cited by 44 (12 self)
 Add to MetaCart
At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topological graph theory to decompose a graph into the structure guaranteed by the theorem: a cliquesum of pieces almostembeddable into boundedgenus surfaces. This result has many applications. In particular, we show applications to developing many approximation algorithms, including a 2approximation to graph coloring, constantfactor approximations to treewidth and the largest grid minor, combinatorial polylogarithmicapproximation to halfintegral multicommodity flow, subexponential fixedparameter algorithms, and PTASs for many minimization and maximization problems, on graphs excluding a fixed minor. 1.
Kernelization Algorithms for the Vertex Cover Problem: Theory and Experiments (Extended Abstract)
"... Faisal N. AbuKhzam + , Rebecca L. Collins + , Michael R. Fellows # , Michael A. Langston + , W. Henry Suters + and Chris T. Symons + 1 ..."
Abstract

Cited by 41 (14 self)
 Add to MetaCart
Faisal N. AbuKhzam + , Rebecca L. Collins + , Michael R. Fellows # , Michael A. Langston + , W. Henry Suters + and Chris T. Symons + 1
Computing crossing numbers in quadratic time
 J. Comput. Syst. Sci
, 2004
"... We show that for every fixed k ≥ 0 there is a quadratic time algorithm that decides whether a given graph has crossing number at most k and, if this is the case, computes a drawing of the graph in the plane with at most k crossings. 1. ..."
Abstract

Cited by 28 (0 self)
 Add to MetaCart
We show that for every fixed k ≥ 0 there is a quadratic time algorithm that decides whether a given graph has crossing number at most k and, if this is the case, computes a drawing of the graph in the plane with at most k crossings. 1.
On Interval Routing Schemes and Treewidth
 and treewidth,inProceedings 21thInternationalWorkshoponGraphTheoreticConceptsinComputerScienceWG'95,M.Nagl,ed.,SpringerVerlag,LectureNotesin ComputerScience,vol.1017,1995,pp.181{186
, 1997
"... In this paper, we investigate which processor networks allow k label Interval Routing Schemes, under the assumption that costs of edges may vary. We show that for each fixed k 1, the class of graphs allowing such routing schemes is closed under minortaking in the domain of connected graphs, and he ..."
Abstract

Cited by 16 (8 self)
 Add to MetaCart
In this paper, we investigate which processor networks allow k label Interval Routing Schemes, under the assumption that costs of edges may vary. We show that for each fixed k 1, the class of graphs allowing such routing schemes is closed under minortaking in the domain of connected graphs, and hence has a linear time recognition algorithm. This result connects the theory of compact routing with the theory of graph minors and treewidth. We show that every graph that does not contain K 2;r as a minor has treewidth at most 2r \Gamma 2. In case the graph is planar, this bound can be lowered to r + 2. As a consequence, graphs that allow klabel Interval Routing Schemes under dynamic cost edges have treewidth at most 4k, and treewidth at most 2k + 3 if they are planar. Similar results are shown for other types of Interval Routing Schemes.
Genomescale computational approaches to memoryintensive applications in systems biology
 In Proceedings, Supercomputing
, 2005
"... Graphtheoretical approaches to biological network analysis have proven to be effective for small networks but are computationally infeasible for comprehensive genomescale systemslevel elucidation of these networks. The difficulty lies in the NPhard nature of many global systems biology problems ..."
Abstract

Cited by 13 (5 self)
 Add to MetaCart
Graphtheoretical approaches to biological network analysis have proven to be effective for small networks but are computationally infeasible for comprehensive genomescale systemslevel elucidation of these networks. The difficulty lies in the NPhard nature of many global systems biology problems that, in practice, translates to exponential (or worse) run times for finding exact optimal solutions. Moreover, these problems, especially those of an enumerative flavor, are often memoryintensive and must share very large sets of data effectively across many processors. For example, the enumeration of maximal cliques – a core component in gene expression networks analysis, cis regulatory motif finding, and the study of quantitative trait loci for highthroughput molecular phenotypes – can result in as many as 3 n/3 maximal cliques for a graph with n vertices. Memory requirements to store those cliques reach terabyte scales even on modestsized genomes. Emerging hardware architectures with ultralarge globally addressable memory such as the SGI Altix and Cray X1 seem to be well suited for addressing these types of dataintensive problems in systems biology. This paper presents a novel framework that provides exact, parallel and scalable solutions to various graphtheoretical approaches to genomescale elucidation of biological networks. This framework takes advantage of these largememory architectures by creating globally addressable bitmap memory indices with potentially high compression rates, fast bitwiselogical operations, and
Fast FixedParameter Tractable Algorithms for Nontrivial Generalizations of Vertex Cover
, 2003
"... 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 ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
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,...