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376
A Partial KArboretum of Graphs With Bounded Treewidth
 J. Algorithms
, 1998
"... The notion of treewidth has seen to be a powerful vehicle for many graph algorithmic studies. This survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class. Also, some mutual relations between such classes ..."
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Cited by 325 (35 self)
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The notion of treewidth has seen to be a powerful vehicle for many graph algorithmic studies. This survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class. Also, some mutual relations between such classes are discussed.
Approximation schemes for covering and packing problems in image processing and VLSI
 J. ACM
, 1985
"... A unified and powerful approach is presented for devising polynomial approximation schemes for many strongly NPcomplete problems. Such schemes consist of families of approximation algorithms for each desired performance bound on the relative error c> 0, with running time that is polynomial whe ..."
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Cited by 249 (0 self)
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A unified and powerful approach is presented for devising polynomial approximation schemes for many strongly NPcomplete problems. Such schemes consist of families of approximation algorithms for each desired performance bound on the relative error c> 0, with running time that is polynomial when c is fixed. Thougb the polynomiality of these algorithms depends on the degree of approximation e being fixed, they cannot be improved, owing to a negative result stating that there are no fully polynomial approximation schemes for strongly NPcomplete problems unless NP = P. The unified technique that is introduced here, referred to as the shifting strategy, is applicable to numerous geometric covering and packing problems. The method of using the technique and how it varies with problem parameters are illustrated. A similar technique, independently devised by B. S. Baker, was shown to be applicable for covering and packing problems on planar graphs.
The NPcompleteness column: an ongoing guide
 JOURNAL OF ALGORITHMS
, 1987
"... 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. Freem ..."
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Cited by 239 (0 self)
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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
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 151 (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 150 (2 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.
On the Complexity of Scheduling in Wireless Networks
 MOBICOM '06
, 2006
"... We consider the problem of throughputoptimal scheduling in wireless networks subject to interference constraints. We model the interference using a family of Khop interference models. We define a Khop interference model as one for which no two links within K hops can successfully transmit at the ..."
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Cited by 129 (3 self)
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We consider the problem of throughputoptimal scheduling in wireless networks subject to interference constraints. We model the interference using a family of Khop interference models. We define a Khop interference model as one for which no two links within K hops can successfully transmit at the same time (Note that IEEE 802.11 DCF corresponds to a 2hop interference model.). For a given K, a throughputoptimal scheduler needs to solve a maximum weighted matching problem subject to the Khop interference constraints. For K = 1, the resulting problem is the classical Maximum Weighted Matching problem, that can be solved in polynomial time. However, we show that for K> 1, the resulting problems are NPHard and cannot be approximated within a factor that grows polynomially with the number of nodes. Interestingly, we show that for specific kinds of graphs, that can be used to model the underlying connectivity graph of a wide range of wireless networks, the resulting problems admit polynomial time approximation schemes. We also show that a simple greedy matching algorithm provides a constant factor approximation to the scheduling problem for all K in this case. We then show that under a setting with singlehop traffic and no rate control, the maximal scheduling policy considered in recent related works can achieve a constant fraction of the capacity region for networks whose connectivity graph can be represented using one of the above classes of graphs. These results are encouraging as they suggest that one can develop distributed algorithms to achieve near optimal throughput in case of a wide range of wireless networks.
NCApproximation Schemes for NP and PSPACEHard Problems for Geometric Graphs
, 1997
"... We present NC approximation schemes for a number of graph problems when restricted to geometric graphs including unit disk graphs and graphs drawn in a civilized manner. Our approximation schemes exhibit the same time versus performance tradeoff as the best known approximation schemes for planar gr ..."
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Cited by 116 (1 self)
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We present NC approximation schemes for a number of graph problems when restricted to geometric graphs including unit disk graphs and graphs drawn in a civilized manner. Our approximation schemes exhibit the same time versus performance tradeoff as the best known approximation schemes for planar graphs. We also define the concept of precision unit disk graphs and show that for such graphs the approximation schemes have a better time versus performance tradeoff than the approximation schemes for arbitrary unit disk graphs. Moreover, compared to unit disk graphs, we show that for precision unit disk graphs, many more graph problems have efficient approximation schemes. Our NC approximation schemes can also be extended to obtain efficient NC approximation schemes for several PSPACEhard problems on unit disk graphs specified using a restricted version of the hierarchical specification language of Bentley, Ottmann and Widmayer. The approximation schemes for hierarchically specified un...
Fixed Parameter Algorithms for Dominating Set and Related Problems on Planar Graphs
, 2002
"... We present an algorithm that constructively produces a solution to the kdominating set problem for planar graphs in time O(c . To obtain this result, we show that the treewidth of a planar graph with domination number (G) is O( (G)), and that such a tree decomposition can be found in O( (G)n) time. ..."
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Cited by 112 (22 self)
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We present an algorithm that constructively produces a solution to the kdominating set problem for planar graphs in time O(c . To obtain this result, we show that the treewidth of a planar graph with domination number (G) is O( (G)), and that such a tree decomposition can be found in O( (G)n) time. The same technique can be used to show that the kface cover problem ( find a size k set of faces that cover all vertices of a given plane graph) can be solved in O(c n) time, where c 1 = 3 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of kdominating set, e.g., kindependent dominating set and kweighted dominating set.
Diameter and Treewidth in MinorClosed Graph Families
, 1999
"... It is known that any planar graph with diameter D has treewidth O(D), and this fact has been used as the basis for several planar graph algorithms. We investigate the extent to which similar relations hold in other graph families. We show that treewidth is bounded by a function of the diameter in a ..."
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Cited by 111 (2 self)
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It is known that any planar graph with diameter D has treewidth O(D), and this fact has been used as the basis for several planar graph algorithms. We investigate the extent to which similar relations hold in other graph families. We show that treewidth is bounded by a function of the diameter in a minorclosed family, if and only if some apex graph does not belong to the family. In particular, the O(D) bound above can be extended to boundedgenus graphs. As a consequence, we extend several approximation algorithms and exact subgraph isomorphism algorithms from planar graphs to other graph families.
PolynomialTime Approximation Schemes for Geometric Graphs
, 2001
"... A disk graph is the intersection graph of a set of disks with arbitrary diameters in the plane. For the case that the disk representation is given, we present polynomialtime approximation schemes (PTASs) for the maximum weight independent set problem (selecting disjoint disks of maximum total weigh ..."
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Cited by 102 (5 self)
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A disk graph is the intersection graph of a set of disks with arbitrary diameters in the plane. For the case that the disk representation is given, we present polynomialtime approximation schemes (PTASs) for the maximum weight independent set problem (selecting disjoint disks of maximum total weight) and for the minimum weight vertex cover problem in disk graphs. These are the first known PTASs for NPhard optimization problems on disk graphs. They are based on a novel recursive subdivision of the plane that allows applying a shifting strategy on different levels simultaneously, so that a dynamic programming approach becomes feasible. The PTASs for disk graphs represent a common generalization of previous results for planar graphs and unit disk graphs. They can be extended to intersections graphs of other "disklike" geometric objects (such as squares or regular polygons), also in higher dimensions.