Results 1  10
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17
Subgraph Isomorphism in Planar Graphs and Related Problems
, 1999
"... We solve the subgraph isomorphism problem in planar graphs in linear time, for any pattern of constant size. Our results are based on a technique of partitioning the planar graph into pieces of small treewidth, and applying dynamic programming within each piece. The same methods can be used to ..."
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Cited by 109 (3 self)
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We solve the subgraph isomorphism problem in planar graphs in linear time, for any pattern of constant size. Our results are based on a technique of partitioning the planar graph into pieces of small treewidth, and applying dynamic programming within each piece. The same methods can be used to solve other planar graph problems including connectivity, diameter, girth, induced subgraph isomorphism, and shortest paths.
Finding and counting given length cycles
 Algorithmica
, 1997
"... We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected graphs. Most of the bounds obtained depend solely on the number of edges in the graph in question, and not on the number of vertices. The bounds obtained improve upon various previ ..."
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Cited by 82 (11 self)
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We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected graphs. Most of the bounds obtained depend solely on the number of edges in the graph in question, and not on the number of vertices. The bounds obtained improve upon various previously known results. 1
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 ..."
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Cited by 67 (15 self)
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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 Complexity for the Skeptic
 In Proc. 18th IEEE Annual Conference on Computational Complexity
, 2003
"... The goal of this article is to provide a tourist guide, with an eye towards structural issues, to what I consider some of the major highlights of parameterized complexity. ..."
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Cited by 36 (1 self)
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The goal of this article is to provide a tourist guide, with an eye towards structural issues, to what I consider some of the major highlights of parameterized complexity.
Splitters and nearoptimal derandomization
"... We present a fairly general method for finding deterministic constructions obeying what we call krestrictions; this yields structures of size not much larger than the probabilistic bound. The structures constructed by our method include (n; k)universal sets (a collection of binary vectors of lengt ..."
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Cited by 34 (1 self)
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We present a fairly general method for finding deterministic constructions obeying what we call krestrictions; this yields structures of size not much larger than the probabilistic bound. The structures constructed by our method include (n; k)universal sets (a collection of binary vectors of length n such that for any subset of size k of the indices, all 2k configurations appear) and families of perfect hash functions. The nearoptimal constructions of these objects imply the very efficient derandomization of algorithms in learning, of fixedsubgraph finding algorithms, and of near optimal threshold formulae. In addition, they derandomize the reduction showing the hardness of approximation of set cover. They also yield deterministic constructions for a localcoloring protocol, and for exhaustive testing of circuits.
Approximating the Longest Cycle Problem in Sparse Graphs
, 2002
"... We consider the problem of finding long paths and cycles in Hamiltonian graphs. The focus of our work is on sparse graphs, e.g., cubic graphs, that satisfy some property known to hold for Hamiltonian graphs, e.g., kcyclability. We first consider the problem of finding long cycles in 3connected cub ..."
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Cited by 11 (2 self)
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We consider the problem of finding long paths and cycles in Hamiltonian graphs. The focus of our work is on sparse graphs, e.g., cubic graphs, that satisfy some property known to hold for Hamiltonian graphs, e.g., kcyclability. We first consider the problem of finding long cycles in 3connected cubic graphs whose edges have weights wi * 0. We find cycles of weight at least (P wai) 1 a for a = log2 3. Based on this result, we develop an algorithm for finding a cycle of length at least m(log3 2)=2 in 3cyclable graphs with vertices of degree at most 3. As a corollary of this result, for arbitrary graphs with vertices of degree at most 3 that have a cycle of length l (or more generally a 3cyclable minor with degrees at most 3 and with l edges), we find a cycle of length at least l(log3 2)=2. We consider the graph property of 1toughness that is common to Hamiltonian graphs and 3 connected cubic graphs, and try to determine if 1toughness implies the existence of long cycles. We show that 2connectivity and 1toughness, for constant degree graphs, may give cycles that are only of logarithmic length. However, we exhibit a class of 3connected 1tough graphs with degrees up to 6 where we can find cycles of length at least mlog3 2=2.
Computational Tractability: The View From Mars
 Bulletin of the European Association of Theoretical Computer Science
"... We describe a point of view about the parameterized computational complexity framework in the broad context of one of the central issues of theoretical computer science as a field: the problem of systematically coping with computational intractability. Those already familiar with the basic ideas of ..."
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Cited by 9 (1 self)
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We describe a point of view about the parameterized computational complexity framework in the broad context of one of the central issues of theoretical computer science as a field: the problem of systematically coping with computational intractability. Those already familiar with the basic ideas of parameterized complexity will nevertheless find here something new: the emerging systematic connections between fixedparameter tractability techniques and the design of useful heuristic algorithms, and also perhaps the philosophical maturation of the parameterized complexity program.
On the Approximation of Finding A(nother) Hamiltonian Cycle in Cubic Hamiltonian Graphs
 J. Algorithms
, 1999
"... It is a simple fact that cubic Hamiltonian graphs have at least two Hamiltonian cycles. Finding such a cycle is NPhard in general, and no polynomialtime algorithm is known for the problem of finding a second Hamiltonian cycle when one such cycle is given as part of the input. We investigate the ..."
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Cited by 8 (0 self)
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It is a simple fact that cubic Hamiltonian graphs have at least two Hamiltonian cycles. Finding such a cycle is NPhard in general, and no polynomialtime algorithm is known for the problem of finding a second Hamiltonian cycle when one such cycle is given as part of the input. We investigate the complexity of approximating this problem where by a feasible solution we mean a(nother) cycle in the graph, and the quality of the solution is measured by cycle length.
Complexity of Searching an Immobile Hider in a Graph
, 1995
"... . We study the computational complexity of certain searchhide games on a graph. There are two players, called searcher and hider. The hider is immobile and hides in one of the nodes of the graph. The searcher selects a starting node and a search path of length at most k. His objective is to dete ..."
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Cited by 6 (0 self)
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. We study the computational complexity of certain searchhide games on a graph. There are two players, called searcher and hider. The hider is immobile and hides in one of the nodes of the graph. The searcher selects a starting node and a search path of length at most k. His objective is to detect the hider, which he does with certainty if he visits the node chosen for hiding. Finding the optimal randomized strategies in this zerosum game defines a fractional path covering problem and its dual, a fractional packing problem. If the length k of the search path is arbitrary, then the problem is NPhard. The problem remains NPhard if the searcher may freely revisit nodes that he has seen before. In that case, the searcher selects a connected subgraph of k nodes rather than a path of k nodes. If k is logarithmic in the number of nodes of the graph, then the problem can be solved in polynomial time; this is shown using a recent technique called colorcoding due to Alon, Yuster...
Some Parameterized Problems on Digraphs
 The Computer Journal
"... We survey known results on parameterized complexity of the feedback set and induced subdigraph problems for digraphs. We prove new results on some parameterizations of the paired comparison problems on digraphs. One of our theorems implies a new result for a parameterized version of the linear arran ..."
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Cited by 6 (2 self)
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We survey known results on parameterized complexity of the feedback set and induced subdigraph problems for digraphs. We prove new results on some parameterizations of the paired comparison problems on digraphs. One of our theorems implies a new result for a parameterized version of the linear arrangement problem for undirected graphs. We state several open problems. 1