M. Goldberg and T. Spencer. An efficient parallel algorithm that finds independent sets of guaranteed size. SIAM J. Disc. Math., 6:443--459, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....hypergraphs with any k 2. We also aim to find large independent sets in non uniform hypergraphs. 1.1 Related Work The following parallel algorithms are known in the case of graphs. Spencer [23] gave an RNC algorithm that yields an IS of expected size ff 2(G) in graphs G. Goldberg and Spencer [10] presented an NC algorithm that finds an IS of size at least dn 2 = 2m n)e in any graph G, where n and m denote the number of vertices and edges in G respectively. This bound equals ff 2 (G) when G is regular; in all other cases, ff 2(G) is higher. In fact, it is not hard to construct graph ....

M. Goldberg and T. Spencer. An efficient parallel algorithm that finds independent sets of guaranteed size. SIAM J. Disc. Math., 6:443--459, 1993.

....minimum degree in G. b) Add v to IS(G) c) Delete v and its neighbors from G. until V = 3. Output IS(G) Lemma 7.5 8ffi 0, 9c 0 0 such that given any ffi near planar graph G(V; E) GREEDY finds an independent set IS(G) in G such that jIS(G)j c 0 jV j. Proof: By Turan s Theorem [GS90], every graph with n nodes and m edges has an independent set of size at least n 2 = 2m n) Further, it is known that GREEDY produces an independent set of at least this size [GS90] Since G is ffi near planar, by Proposition 7.4, m cn for some constant c. Thus, jIS(G)j n= 2c 1) that is, ....

....G(V; E) GREEDY finds an independent set IS(G) in G such that jIS(G)j c 0 jV j. Proof: By Turan s Theorem [GS90] every graph with n nodes and m edges has an independent set of size at least n 2 = 2m n) Further, it is known that GREEDY produces an independent set of at least this size [GS90]. Since G is ffi near planar, by Proposition 7.4, m cn for some constant c. Thus, jIS(G)j n= 2c 1) that is, the lemma holds with c 0 = 1= 2c 1) We now prove the main theorem of this section. Theorem 7.6 For any fixed ffi 0, there are NC approximation schemes for the following MAX ....

M. Goldberg and T. H. Spencer, "An Efficient Parallel Algorithm that Finds Independent Sets of Guaranteed Size", Proc. First Annual ACM-SIAM Symp. on Discrete Algorithms (SODA'90), San Francisco, CA, 1990, pp. 219--225.

....I V is called independent, if I contains no edges. The independence number ff(G) is the size of a largest independent set. Computing an independent set I with jI j = ff(G) is an NP hard problem. Therefore, polynomial time approximation algorithms for this problem have been investigated, cf. 1] [9]. For an optimization instance I , let OPT (I) denote the value of the optimal solution and A(I) the solution found by an algorithm A. The approximation ratio AR(n) of algorithm A is defined by max In ae OPT (I n ) A(I n ) A(I n ) OPT (I n ) oe where the maximum is taken over all ....

M. Goldberg and T. Spencer, An Efficient Parallel Algorithm that Finds Independent Sets of Guaranteed Size, SIAM Journal of Discrete Mathematics 6, 1993, 443-459.

....was known. In some cases, the corresponding upper bounds match the lower bounds up to constant factors. The involved concepts are uncrowded hypergraphs. 1 Introduction A fundamental problem in Computer Science and Mathematics is to find a large independent set in an arbitrary graph [6] 14] [20]. Recall, that for a graph G = V; E) with vertex set V and edgeset E [V ] 2 a subset I V of the vertex set is called independent, if the subgraph induced on I contains no edges e 2 E, i.e. E [I] 2 = The maximum cardinality of an independent set I is called the independence number ....

M. Goldberg and T. Spencer, An Efficient Parallel Algorithm that Finds Independent Sets of Guaranteed Size, SIAM Journal of Discrete Mathematics 6, 1993, 443-459.

....upper bound on the chromatic sum has been observed several times in the past. Let m denote the number of edges in the graph. Lemma 4.2 ( 9, 23] The sum of any compact coloring is at most m n. This bound is tight for disjoint collection of cliques. It can be attained by a parallel algorithm [17]. Theorem 4.3 Any compact coloring of a graph G = V; E) provides a Delta 2 3 approximation to MCS(G) and that is tight. Proof: All edges have at least one endpoint outside the first color class of the optimal solution. Thus, when maximum degree is bounded by Delta, there are at least ....

Goldberg, M. and Spencer, T. H. (1990), An efficient parallel algorithm that finds independent sets of guaranteed size, in Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 219--225.

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