Parallel algorithms for fractional and maximal independent sets in planar graphs

Dadoun, Norm; Kirkpatrick, David G.

doi:10.1016/0166-218x(90)90130-5

Cited by 4 publications

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“…Returning now to MAXLVP, let us note that a more efficient NC algorithm for the special case when G is planar was found by Xin He [206] 3 -free graphs which we will meet again in the final section of this paper.…”

Section: Vertex Packing In Parallelmentioning

confidence: 99%

“…Returning now to MAXLVP, let us note that a more efficient NC algorithm for the special case when G is planar was found by Xin He [206] who found an NC 2 routine requiring only 0(n) processors. In 1988, this result was extended by Khuller [207] to K 3 , 3 -free graphs.…”

Section: Vertex Packing In Parallelmentioning

confidence: 99%

“…Can such a set be found via an NC algorithm? In [205], they answered their own question in the affirmative, giving an NC 3 algorithm.Returning now to MAXLVP, let us note that a more efficient NC algorithm for the special case when G is planar was found by Xin He [206] who found an NC 2 routine requiring only 0(n) processors. In 1988, this result was extended by Khuller [207] to K 3 , 3 -free graphs.…”

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Matching and Vertex Packing: How Hard Are They?

Plummer¹

1991

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ECE 21 ABSTACTFEB 19920Two of the most well-known problems in graph theory are: D (a) Find a maximum matching (or perfect matching, if one exists), and (b) Find a maximum independent set of vertices.The first problem-usually called the matching problem-is known to have a polynomial algorithm; the second-often called the vertex packing problem-is known to be NP-complete. However, many graph theorists-especially those who do not deal much with complexity of algorithms-know little more about the complexity issues associated with these two problems than these two basic facts.What is not so widely known within the graph theory community is that these two problems have motivated a great deal of recent activity in the area of algorithms and their complexity. Of course it is not known whether or not P = NP, but most workers in the area currently believe that equality is unlikely to hold. Motivated by this belief, a number of people have studied variations of both matching and vertex packing with the general theme being two-fold. On the one hand, one can add various side conditions to the matching problem and study the complexity-both sequential and parallel-of the resulting --problems. On the other hand, one can investigate certain large and interesting classes of graphs trying to prove that for these classes the vertex packing problem has a polynomial solution.Each branch of this two-pronged attack has yielded both interesting theorems and 0 perplexing unsolved problems. This paper will survey this work. I Introduction and Background: The two fundamental ProblemsIn this paper, graphs will be assumed to be connected, undirected and will have no multiple edges or loops. A matching is any set of independent edges; i.e., no two have a vertex in common. A maximal matching is a matching not properly contained in any other matching. A maximum matching is one of largest cardinality. A perfect matching (sometimes called a 1-factor) in a graph G is a matching which covers all vertices of G. A set of vertices S C V(G) is independent if no two vertices of S are adjacent. An * work supported by ONR Contract #N00014-85-K-0488, #N00014-91-J-1142 and Laboratoire de Recherche en Informatique, CNRS, Univ. Paris Sud ___ 1 ir P'zblic release and sale; its ) I di-ziutIon is unlimited. independent set S is maximal if it is not a proper subset of any other independent set and maximum if it is an independent set of largest cardinality.So far, then, maximal and maximum matchings and independent sets are quite analogous; matchings corresponding to sets of edges and independent sets corresponding to sets of vertices. To be sure, they are related. A (maximal, maximum) matching in a graph G corresponds to a (maximal, maximum) independent set in the line graph L(G). But we shall soon see that the concepts quickly diverge in difficulty.First, however, let us note that matching and vertex packing remain closely related, at least in the computational sense, if one considers the class of bipartite graphs. Historically speaking, it was this class of gra...

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Section: Vertex Packing In Parallelmentioning

confidence: 99%

Section: Vertex Packing In Parallelmentioning

confidence: 99%

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confidence: 99%

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Matching and Vertex Packing: How Hard Are They?

Plummer¹

1991

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“…Such independent sets can be created in parallel. Dadoun and Kirkpatric [5] and Karp and Wigderson [15] presented algorithms that find independent set in polylogarithmic time (assuming a very high number of processors).…”

Section: Related Workmentioning

confidence: 99%

Simplification of FEM-Models on Cell BE

Hjelmervik

Léon

2010

Mathematical Methods for Curves and Surfaces

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Abstract. Preparing a CAD model for Finite Element (FE) analysis can be a time-consuming task, where shape and mesh simplifications play an important role. It is important that the simplified model has the same mechanical properties as the original one, and that the deviation from the original stays within a given tolerance. Most FE mesh simplification algorithms are either fully or partially sequential, and are therefore not suitable for architectures with high levels of parallelism. Furthermore, the use of processors such as GPUs of IBMs Cell BE requires algorithms to be adapted to benefit from their computational advantages. Here, we present an algorithm written for parallel processors, and its implementation for the Cell BE.

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Optimal parallel 3-coloring algorithm for rooted trees and its applications

Rajčáni

1992

Information Processing Letters

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Parallel algorithms for fractional and maximal independent sets in planar graphs

Cited by 4 publications

References 11 publications

Matching and Vertex Packing: How Hard Are They?

Matching and Vertex Packing: How Hard Are They?

Simplification of FEM-Models on Cell BE

Optimal parallel 3-coloring algorithm for rooted trees and its applications

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