M. Krivelevich. Approximate set covering in uniform hypergraphs. J. Algorithms, 25(1):118--143, October 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....we consider k uniform hypergraphs, which are hypergraphs in which all edges are of size k. All our results can be easily extended to the case where each edge is of size at most k. A k approximation algorithm is straightforward using a maximal matching. For any fixed k, the best algorithm known [21] achieves a ratio of k(1 Gamma c=n k Gamma1 k ) For bounded degree k uniform hypergraphs, the best algorithm known [21] achieves a ratio of k(1 Gamma c= Delta 1 k Gamma1 ) In this paper we present an algorithm which, for a fixed k, achieves a ratio of (k Gamma (1 Gamma o(1) k(k Gamma ....

....extended to the case where each edge is of size at most k. A k approximation algorithm is straightforward using a maximal matching. For any fixed k, the best algorithm known [21] achieves a ratio of k(1 Gamma c=n k Gamma1 k ) For bounded degree k uniform hypergraphs, the best algorithm known [21] achieves a ratio of k(1 Gamma c= Delta 1 k Gamma1 ) In this paper we present an algorithm which, for a fixed k, achieves a ratio of (k Gamma (1 Gamma o(1) k(k Gamma 1) ln ln Delta ln Delta ) for k uniform hypergraphs, where Delta is the maximal degree in the hypergraph. This result ....

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M. Krivelevich. Approximate set covering in uniform hypergraphs. J. Algorithms, 25(1):118--143, October 1997.

....of rank at most r, then: H) H) r 1 : This bound is tight. Theorem 4. Let 2 r k be integers. If H is a k colourable hypergraph of rank at most r, then: H) H) k 1 k r : This bound is tight. These results are described in Section 3. In a subsequent paper [2] the above two theorems are applied to the design of approximation algorithms for the set covering problem. A strong k colouring of the hypergraph H = V; E) is a partition (C 1 ; C k ) of the set of vertices V into k classes (colours) such that no colour appears more than once in the same ....

M. Krivelevich, Approximate set covering in uniform hypergraphs, preprint.

....Ratio Approach The so called Local Ratio Approach was pioneered by Bar Yehuda and Even [4] who used it to develop an approximate graph vertex cover algorithm. Boppana and Halld orsson [6] used the Local Ratio Approach in their algorithm for approximating the independence number of a graph. In [18], this approach has been applied to a hypergraph approximation problem. Here we use the Local Ratio Lemma to improve the result of Theorem 1 for certain values of the parameter . Our approach is similar in spirit to that of [6] Notation. For a hypergraph H = V; E) the local ratio of H is de ....

M. Krivelevich, Approximate set covering in uniform hypergraphs, J. Algorithms 25 (1997), 118{ 143.

....(H) r Gamma 1 : This bound is tight. COVERS IN r PARTITE HYPERGRAPHS 5 Theorem 4. Let 2 r k be integers. If H is a k colourable hypergraph of rank at most r, then: H) H) k Gamma 1 k r : This bound is tight. These results are described in Section 3. In a subsequent paper [2] the above two theorems are applied to the design of approximation algorithms for the set covering problem. A strong k colouring of the hypergraph H = V; E) is a partition (C 1 ; C k ) of the set of vertices V into k classes (colours) such that no colour appears more than once in the same ....

M. Krivelevich, Approximate set covering in uniform hypergraphs, preprint.

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