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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 .
Approximation Algorithms for Classes of Graphs Excluding SingleCrossing Graphs as Minors
"... Many problems that are intractable for general graphs allow polynomialtime solutions for structured classes of graphs, such as planar graphs and graphs of bounded treewidth. ..."
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Cited by 25 (16 self)
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Many problems that are intractable for general graphs allow polynomialtime solutions for structured classes of graphs, such as planar graphs and graphs of bounded treewidth.
Multicoloring Planar Graphs and Partial kTrees
 In Proceedings of the Second International Workshop on Approximation algorithms (APPROX '99). Lecture Notes in Computer Science
, 2000
"... We study graph multicoloring problems, motivated by the scheduling of dependent jobs on multiple machines. In multicoloring problems, vertices have lengths which determine the number of colors they must receive, and the desired coloring can be either contiguous (nonpreemptive schedule) or arbitr ..."
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Cited by 5 (3 self)
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We study graph multicoloring problems, motivated by the scheduling of dependent jobs on multiple machines. In multicoloring problems, vertices have lengths which determine the number of colors they must receive, and the desired coloring can be either contiguous (nonpreemptive schedule) or arbitrary (preemptive schedule). We consider both the sumofcompletion times measure, or the sum of the last color assigned to each vertex, as well as the more common makespan measure, or the number of colors used. In this paper, we study two fundamental classes of graphs: planar graphs and partial ktrees. For both classes, we give a polynomial time approximation scheme (PTAS) for the multicoloring sum, for both the preemptive and nonpreemptive cases. On the other hand, we show the problem to be strongly NPhard on planar graphs, even in the unweighted case, known as the Sum Coloring problem. For nonpreemptive multicoloring sum of partial ktrees, we obtain an fully polynomial time app...
Fast approximation schemes for K_{3,3}minorfree or K_5minorfree graphs
"... As the class of graphs of bounded treewidth is of limited size, we need to solve NPhard problems for wider classes of graphs than this class. Eppstein introduced a new concept which can be considered as a generalization of bounded treewidth. A graph G has locally bounded treewidth if for each verte ..."
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Cited by 5 (5 self)
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As the class of graphs of bounded treewidth is of limited size, we need to solve NPhard problems for wider classes of graphs than this class. Eppstein introduced a new concept which can be considered as a generalization of bounded treewidth. A graph G has locally bounded treewidth if for each vertex v of G, the treewidth of the subgraph of G induced on all vertices of distance at most r from v is only a function of r, called local treewidth. So far the only graphs determined to have small local treewidth are planar graphs. In this paper, we prove that any graph excluding one of K5 or K3;3 as a minor has local treewidth bounded by 3k + 4. As a result, we can design practical polynomialtime approximation schemes for both minimization and maximization problems on these classes of nonplanar graphs.
Algorithms for Graphs of (Locally) Bounded Treewidth
, 2001
"... Many reallife problems can be modeled by graphtheoretic problems. These graph problems are usually NPhard and hence there is no efficient algorithm for solving them, unless P= NP. One way to overcome this hardness is to solve the problems when restricted to special graphs. Trees are one kind of g ..."
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Cited by 4 (3 self)
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Many reallife problems can be modeled by graphtheoretic problems. These graph problems are usually NPhard and hence there is no efficient algorithm for solving them, unless P= NP. One way to overcome this hardness is to solve the problems when restricted to special graphs. Trees are one kind of graph for which several NPcomplete problems can be solved in polynomial time. Graphs of bounded treewidth, which generalize trees, show good algorithmic properties similar to those of trees. Using ideas developed for tree algorithms, Arnborg and Proskurowski introduced a general dynamic programming approach which solves many problems such as dominating set, vertex cover and independent set. Others used this approach to solve other NPhard problems. Matousek and Thomas applied this approach to solve the subgraph isomorphism problem when the source graph has bounded degree and the host graph has bounded treewidth. In this thesis, we introduce a new property for graphs called logbounded fragmentation, by which we mean after removing any set of at most k vertices the number of connected components is at most O(k log n), where n is the number of vertices of the graph. We then extend the result of Matousek and Thomas to the case in which the source graph is a logbounded fragmentation graph and the host graph has bounded treewidth. Besides this result, we demonstrate how bounded fragmentation might be used to measure the reliability of a network.
Tools for Multicoloring with Applications to Planar Graphs and Partial kTrees ∗
, 2001
"... We study graph multicoloring problems, motivated by the scheduling of dependent jobs on multiple machines. In multicoloring problems, vertices have lengths which determine the number of colors they must receive, and the desired coloring can be either contiguous (nonpreemptive schedule) or arbitrary ..."
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We study graph multicoloring problems, motivated by the scheduling of dependent jobs on multiple machines. In multicoloring problems, vertices have lengths which determine the number of colors they must receive, and the desired coloring can be either contiguous (nonpreemptive schedule) or arbitrary (preemptive schedule). We consider both the sumofcompletion times measure, or the sum of the last color assigned to each vertex, as well as the more common makespan measure, or the number of colors used. In this paper, we study two fundamental classes of graphs: planar graphs and partial ktrees. For both classes, we give a polynomial time approximation scheme (PTAS) for the multicoloring sum, for both the preemptive and nonpreemptive cases. On the other hand, we show the problem to be strongly NPhard on planar graphs, even in the unweighted case, known as the Sum Coloring problem. For nonpreemptive multicoloring sum of partial ktrees, we obtain a fully polynomial time approximation scheme. This is based on a pseudopolynomial time algorithm that holds for a general class of cost functions. Finally, we give a PTAS for the makespan of a preemptive multicoloring of partial ktrees that uses only O(logn) preemptions. These results are based on several properties of multicolorings and tools for manipulating them, which may be of more general applicability.
IMPROVED INDUCED MATCHINGS IN SPARSE GRAPHS
, 2009
"... An induced matching in graph G is a matching which is an induced subgraph of G. Clearly, among two vertices with the same neighborhood (called twins) at most one is matched in any induced matching, and if one of them is matched then there is another matching of the same size that matches the other v ..."
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An induced matching in graph G is a matching which is an induced subgraph of G. Clearly, among two vertices with the same neighborhood (called twins) at most one is matched in any induced matching, and if one of them is matched then there is another matching of the same size that matches the other vertex. Motivated by this, Kanj, Pelsmajer, Schaefer and Xia [10] studied induced matchings in twinless graphs. They showed that any twinless planar graph contains an induced matching of size at least n 40 and that there are twinless planar graphs that do not contain an induced matching of size greater than n + O(1). We improve both these bounds to n +O(1), which is tight up to an additive constant. This implies 28 that the problem of deciding an whether a planar graph has an induced matching of size k has a kernel of size at most 28k. We also show for the first time that this problem is FPT for graphs of bounded arboricity. Kanj et al. presented also an algorithm which decides in O(2159√k +n)time whether an nvertex planar graph contains an induced matching of size k. Our results improve the time complexity analysis of their algorithm. However, we show also a more efficient, O(226√k +n)time algorithm based on the branchwidth decomposition. 27 1