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42
Approximation Algorithms for Disjoint Paths Problems
, 1996
"... The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NPcomplete problems for w ..."
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Cited by 140 (0 self)
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The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NPcomplete problems for which very little is known from the point of view of approximation algorithms. It has recently been brought into focus in work on problems such as VLSI layout and routing in highspeed networks; in these settings, the current lack of understanding of the disjoint paths problem is often an obstacle to the design of practical heuristics.
Disjoint Paths in Densely Embedded Graphs
 in Proceedings of the 36th Annual Symposium on Foundations of Computer Science
, 1995
"... We consider the following maximum disjoint paths problem (mdpp). We are given a large network, and pairs of nodes that wish to communicate over paths through the network  the goal is to simultaneously connect as many of these pairs as possible in such a way that no two communication paths share a ..."
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Cited by 60 (6 self)
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We consider the following maximum disjoint paths problem (mdpp). We are given a large network, and pairs of nodes that wish to communicate over paths through the network  the goal is to simultaneously connect as many of these pairs as possible in such a way that no two communication paths share an edge in the network. This classical problem has been brought into focus recently in papers discussing applications to routing in highspeed networks, where the current lack of understanding of the mdpp is an obstacle to the design of practical heuristics. We consider the class of densely embedded, nearlyEulerian graphs, which includes the twodimensional mesh and many other planar and locally planar interconnection networks. We obtain a constantfactor approximation algorithm for the maximum disjoint paths problem for this class of graphs; this improves on an O(log n)approximation for the special case of the twodimensional mesh due to AumannRabani and the authors. For networks that ...
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 ..."
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Cited by 47 (12 self)
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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.
Equivalence of Local Treewidth and Linear Local Treewidth and its Algorithmic Applications
 In Proceedings of the 15th ACMSIAM Symposium on Discrete Algorithms (SODA’04
, 2003
"... We solve an open problem posed by Eppstein in 1995 [14, 15] and reenforced by Grohe [16, 17] concerning locally bounded treewidth in minorclosed families of graphs. A graph has bounded local treewidth if the subgraph induced by vertices within distance r of any vertex has treewidth bounded by a f ..."
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Cited by 28 (11 self)
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We solve an open problem posed by Eppstein in 1995 [14, 15] and reenforced by Grohe [16, 17] concerning locally bounded treewidth in minorclosed families of graphs. A graph has bounded local treewidth if the subgraph induced by vertices within distance r of any vertex has treewidth bounded by a function of r (not n). Eppstein characterized minorclosed families of graphs with bounded local treewidth as precisely minorclosed families that minorexclude an apex graph, where an apex graph has one vertex whose removal leaves a planar graph. In particular, Eppstein showed that all apexminorfree graphs have bounded local treewidth, but his bound is doubly exponential in r, leaving open whether a tighter bound could be obtained. We improve this doubly exponential bound to a linear bound, which is optimal. In particular, any minorclosed graph family with bounded local treewidth has linear local treewidth. Our bound generalizes previously known linear bounds for special classes of graphs proved by several authors. As a consequence of our result, we obtain substantially faster polynomialtime approximation schemes for a broad class of problems in apexminorfree graphs, improving the running time from .
Fast Parameterized Algorithms for Graphs on Surfaces: Linear Kernel and Exponential Speedup
"... Preprocessing by data reduction is a simple but powerful technique used for practically solving di#erent network problems. A number of empirical studies shows that a set of reduction rules for solving Dominating Set problems introduced by Alber, Fellows & Niedermeier leads e#ciently to optimal s ..."
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Cited by 23 (5 self)
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Preprocessing by data reduction is a simple but powerful technique used for practically solving di#erent network problems. A number of empirical studies shows that a set of reduction rules for solving Dominating Set problems introduced by Alber, Fellows & Niedermeier leads e#ciently to optimal solutions for many realistic networks. Despite of the encouraging experiments, the only class of graphs with proven performance guarantee of reductions rules was the class of planar graphs.
An Improved Algorithm For Finding Tree Decompositions Of Small Width
, 2000
"... We present a modification of Bodlaender's linear time algorithm that, for constant k, determines whether an input graph G has treewidth k and, if so, constructs a tree decomposition of G of width at most k. Our algorithm has the following additional feature: if G has treewidth greater than k then a ..."
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Cited by 16 (4 self)
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We present a modification of Bodlaender's linear time algorithm that, for constant k, determines whether an input graph G has treewidth k and, if so, constructs a tree decomposition of G of width at most k. Our algorithm has the following additional feature: if G has treewidth greater than k then a subgraph G^0 of G of treewidth greater than k is returned along with a tree decomposition of G^0 of width at most 2k. A consequence is that the fundamental disjoint rooted paths problem can now be solved in O(n^2) time. This is the primary motivation for this paper.
Graph and map isomorphism and all polyhedral embeddings in linear time
 In Proceedings of the 40th Annual ACM Symposium on Theory of Computing (STOC
, 2008
"... For every surface S (orientable or nonorientable), we give a linear time algorithm to test the graph isomorphism of two graphs, one of which admits an embedding of facewidth at least 3 into S. This improves a previously known algorithm whose time complexity is n O(g),wheregis the genus of S. This ..."
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Cited by 16 (5 self)
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For every surface S (orientable or nonorientable), we give a linear time algorithm to test the graph isomorphism of two graphs, one of which admits an embedding of facewidth at least 3 into S. This improves a previously known algorithm whose time complexity is n O(g),wheregis the genus of S. This is the first algorithm for which the degree of polynomial in the time complexity does not depend on g. The above result is based on two linear time algorithms, each of which solves a problem that is of independent interest. The first of these problems is the following one. Let S beafixedsurface. GivenagraphG andanintegerk≥3, we want to find an embedding of G in S of facewidth at least k, or conclude that such an embedding does not exist. It is known that this problem is NPhard when the surface is not fixed. Moreover, if there is an embedding, the algorithm can give all embeddings of facewidth at least k, up to Whitney equivalence. Here, the facewidth of an embedded graph G is the minimum number of points of G in which some noncontractible closed curve in the surface intersects the graph. In the proof of the above algorithm, we give a simpler proof and a better bound for the theorem by Mohar and Robertson concerning the number of polyhedral embeddings of 3connected graphs.
An algorithm for finding an induced cycle in planar graphs and bounded genus graphs
"... In this paper, we consider the problem of finding an induced cycle passing through k given vertices, which we call the induced cycle problem. The significance of finding induced cycles stems from the fact that precise characterization of perfect graphs would require structures of graphs without an o ..."
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Cited by 12 (0 self)
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In this paper, we consider the problem of finding an induced cycle passing through k given vertices, which we call the induced cycle problem. The significance of finding induced cycles stems from the fact that precise characterization of perfect graphs would require structures of graphs without an odd induced cycle, and its complement. There has been huge progress in the recent years, especially, the Strong Perfect Graph Conjecture was solved in [6]. Concerning recognition of perfect graphs, there had been a longstanding open problem for detecting an odd hole and its complement, and finally this was solved in [4]. Unfortunately, the problem of finding an induced cycle passing through two given vertices is NPcomplete in a general graph [2]. However, if the input graph is constrained to be planar and k is fixed, then the induced cycle problem can be solved in polynomial time [13, 14, 16]. In particular, an O(n 2) time algorithm is given for the case k = 2 by McDiarmid, Reed, Schrijver and Shepherd [18], where n is the number of vertices of the input graph. Our main results in this paper are to improve their result in the following sense. 1. The number of vertices k is allowed to be nontrivially log n super constant number, up to k = o(( log n log log n) 2 3). More precisely, when k = o(( log log n) 2 3), then the ICP in planar graphs can be solved in O(n 2+ε) time for any ε> 0. 2. The time complexity is linear if the given graph is planar and k is fixed. 3. The above results are extended to graphs embedded in a fixed surface. We note that the linear time algorithm (the second result) is independent from the first result. Let us point out that we give the first polynomial time algorithm for the problem for the bounded genus case. In fact, our proof gives a short proof of a result announced in [20] (without complete proof) which gives a linear time algorithm for the disjoint paths problem for fixed k for the bounded genus case. We also extend this result to the induced disjoint paths problem.
The bidimensional theory of boundedgenus graphs
 SIAM Journal on Discrete Mathematics
, 2004
"... 1 Introduction The recent theory of fixedparameter algorithms and parameterized complexity [13] has attracted much attention in its less than 10 years of existence. In general the goal is to understand when NPhard problems have algorithms thatare exponential only in a parameter k of the problem i ..."
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Cited by 11 (6 self)
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1 Introduction The recent theory of fixedparameter algorithms and parameterized complexity [13] has attracted much attention in its less than 10 years of existence. In general the goal is to understand when NPhard problems have algorithms thatare exponential only in a parameter k of the problem instead of the problemsize n. Fixedparameter algorithms whose running time is polynomial for fixedparameter valuesor more precisely
Coloring Locally Bipartite Graphs on Surfaces
, 2000
"... It is proved that there is a function f : N ..."