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14
Simple heuristics for unit disk graphs
 NETWORKS
, 1995
"... Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NPhard optimization problems on unit disk graphs. The problems considered include maximum independent set, minimum vertex cover, minimum coloring ..."
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Cited by 126 (6 self)
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Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NPhard optimization problems on unit disk graphs. The problems considered include maximum independent set, minimum vertex cover, minimum coloring and minimum dominating set. We also present an online coloring heuristic which achieves a competitive ratio of 6 for unit disk graphs. Our heuristics do not need a geometric representation of unit disk graphs. Geometric representations are used only in establishing the performance guarantees of the heuristics. Several of our approximation algorithms can be extended to intersection graphs of circles of arbitrary radii in the plane, intersection graphs of regular polygons, and to intersection graphs of higher dimensional regular objects.
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 113 (1 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.
Planar Orientations with Low OutDegree and Compaction of Adjacency Matrices
 Theoretical Computer Science
, 1991
"... We consider the problem of orienting the edges of a planar graph in such a way that the outdegree of each vertex is minimized. If, for each vertex v, the outdegree is at most d, then we say that such an orientation is dbounded. We prove the following results: ffl Each planar graph has a 5bounde ..."
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Cited by 34 (3 self)
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We consider the problem of orienting the edges of a planar graph in such a way that the outdegree of each vertex is minimized. If, for each vertex v, the outdegree is at most d, then we say that such an orientation is dbounded. We prove the following results: ffl Each planar graph has a 5bounded acyclic orientation, which can be constructed in linear time. ffl Each planar graph has a 3bounded orientation, which can be constructed in linear time. ffl A 6bounded acyclic orientation, and a 3bounded orientation, of each planar graph can each be constructed in parallel time O(log n log n) on an EREW PRAM, using O(n= log n log n) processors. As an application of these results, we present a data structure such that each entry in the adjacency matrix of a planar graph can be looked up in constant time. The data structure uses linear storage, and can be constructed in linear time. Department of Mathematics and Computer Science, University of California, Riverside, CA 92521. On...
Sequential Elimination Graphs
"... A graph is chordal if it does not contain any induced cycle of size greater than three. An alternative characterization of chordal graphs is via a perfect elimination ordering, which is an ordering of the vertices such that, for each vertex v, the neighbors of v that occur later than v in the order ..."
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Cited by 6 (2 self)
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A graph is chordal if it does not contain any induced cycle of size greater than three. An alternative characterization of chordal graphs is via a perfect elimination ordering, which is an ordering of the vertices such that, for each vertex v, the neighbors of v that occur later than v in the ordering form a clique. Akcoglu et al [2] define an extension of chordal graphs whereby the neighbors of v that occur later than v in the elimination order have at most k independent vertices. We refer to such graphs as sequentially kindependent graphs. Clearly this extension of chordal graphs also extends the class of (k+1)clawfree graphs. We study properties of such families of graphs, and we show that several natural classes of graphs are sequentially kindependent for small k. In particular, any intersection graph of translates of a convex object in a two dimensional plane is a sequentially 3independent graph; furthermore, any planar graph is a sequentially 3independent graph. For any fixed constant k, we develop simple, polynomial time approximation algorithms for sequentially kindependent graphs with respect to several wellstudied NPcomplete problems based on this ksequentially independent ordering. Our generalized formulation unifies and extends several previously known results. We also consider other classes of sequential elimination graphs.
Triangle sparsifiers
 Journal of Graph Algorithms and Applications
"... In this work, we introduce the notion of triangle sparsifiers, i.e., sparse graphs which are approximately the same to the original graph with respect to the triangle count. This results in a practical triangle counting method with strong theoretical guarantees. For instance, for unweighted graphs w ..."
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Cited by 5 (1 self)
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In this work, we introduce the notion of triangle sparsifiers, i.e., sparse graphs which are approximately the same to the original graph with respect to the triangle count. This results in a practical triangle counting method with strong theoretical guarantees. For instance, for unweighted graphs we show a randomized algorithm for approximately counting the number of triangles in a graph G, which proceeds as follows: keep each edge independently with probability p, enumerate the triangles in the sparsified graph G ′ and return the number of triangles found in G ′ multiplied by p −3. We prove that under mild assumptions on G and p our algorithm returns a good approximation for the number of triangles with high probability. Specifically, we show that if p ≥ max ( polylog(n)∆ t polylog(n) t1/3), where n, t, ∆, and T denote the number of vertices in G, the number of triangles in G, the maximum number of triangles an edge of G is contained and our triangle count estimate respectively, then T is strongly concentrated around t: Pr [T − t  ≥ ɛt] ≤ n −K. We illustrate the efficiency of our algorithm on various large realworld datasets where we obtain significant speedups. Finally, we investigate cut and spectral sparsifiers with respect to triangle counting and show that they are not optimal. Submitted:
Complexity and approximation results for the connected vertex cover problem
"... Abstract. We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APXhard in bipartite graphs and is 5/3approximable in any class of gr ..."
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Cited by 3 (0 self)
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Abstract. We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APXhard in bipartite graphs and is 5/3approximable in any class of graphs where the vertex cover problem is polynomial (in particular in bipartite graphs).
Optimization problems in unitdisk graphs
 In Christodoulos A. Floudas and Panos M. Pardalos, editors, Encyclopedia of Optimization
, 2009
"... UnitDisk Graphs (UDGs) are intersection graphs of equal diameter (or unit diameter w.l.o.g.) circles in the Euclidean plane. In the geometric (or disk) representation, each ..."
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Cited by 3 (0 self)
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UnitDisk Graphs (UDGs) are intersection graphs of equal diameter (or unit diameter w.l.o.g.) circles in the Euclidean plane. In the geometric (or disk) representation, each
Optimal Placement of Wavelength Converters in Trees and Trees of Rings
, 1999
"... In wavelength routed optical networks, wavelength converters can potentially reduce the requirement on the number of wavelengths. The problem of placing a minimum number of wavelength converters in a WDM network so that any routing can be satisfied using no more wavelengths than if there were wavele ..."
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Cited by 2 (0 self)
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In wavelength routed optical networks, wavelength converters can potentially reduce the requirement on the number of wavelengths. The problem of placing a minimum number of wavelength converters in a WDM network so that any routing can be satisfied using no more wavelengths than if there were wavelength converters at every node was raised in and shown to be NPcomplete in general WDM networks. Recently, it was proved in that this problem is as hard as the wellknown minimum vertex cover problem. In this paper, we further their study in two topologies that are of more practical concrete relevance to the telecommunications industry: trees and tree of rings. We show that the optimal wavelength converter placement problem in these two practical topologies are tractable. Efficient polynomialtime algorithms are presented. Keywords: Wavelength routed optical network, wavelength converter, minimum vertex cover, tree, tree of rings. *This work was supported, in part, by the National Science Foundation under the National Young Investigator Pro gram, contract IRI9357785 and by the Research Institute (IITRI). of Computer Science, Illinois Institute of Technology, Chicago, IL 60616. Department of Computer Science, Illinois Institute of Technology, Chicago, IL 60616. of Computer Science, Illinois Institute of Technology, Chicago, IL 60616.