#### Randomized Approximation for the Set Multicover Problem in Hypergraphs.

El Ouali, M., Munstermann, P. and Srivastav, Anand
(2016)
*Randomized Approximation for the Set Multicover Problem in Hypergraphs. *
Algorithmica, 74
(2).
pp. 574-588.
DOI 10.1007/s00453-014-9962-9.

## Abstract

Let b is an element of N->= 1 and let H = (V, epsilon) be a hypergraph with maximum vertex degree Delta and maximum edge size l. A set b-multicover in H is a set of edges C subset of epsilon such that every vertex in V belongs to at least edges in C SET b-MULTICOVER is the problem of finding a set b-multicover of minimum cardinality, and for b=1 it is the fundamental set cover problem. Peleg et al. (Algorithmica 18(1):44-66, 1997) gave a randomized algorithm achieving an approximation ratio of delta . (1-(c/n)(1/delta)), where delta := Delta-b+1 and c > 0 is a constant. As this ratio depends on the instance size n and tends to delta as n tends to infinity, it remained an open problem whether an approximation ratio of delta alpha with a constant alpha < 1 can be proved. In fact, the authors conjectured that for any fixed Delta and b, the problem is not approximable within a ratio smaller than delta, unless p = NP. We present a randomized algorithm of hybrid type for SET b-MULTICOVER, b >= 2, combining LP-based randomized rounding with greedy repairing, and achieve an approximation ratio of delta. (1- 11(Delta-b)/72l)) for hypergraphs with maximum edge size l is an element of Omicron (max {(nb)(1/5),n(1/4)}) . In particular, for all hypergraphs where l is constant, we get an alpha delta-ratio with constant alpha < 1. Hence the above stated conjecture does not hold for hypergraphs with constant l and we have identified the boundedness of the maximum hyperedge size as a relevant parameter responsible for approximations below delta.

Document Type: | Article |
---|---|

Additional Information: | Times Cited: 0 El Ouali, Mourad Munstermann, Peter Srivastav, Anand |

Keywords: | Combinatorial optimization Approximation algorithms Randomized algorithm Hypergraphs Set cover and set multicover |

Research affiliation: | Kiel University > Kiel Marine Science OceanRep > The Future Ocean - Cluster of Excellence Kiel University |

Refereed: | Yes |

Open Access Journal?: | No |

DOI etc.: | 10.1007/s00453-014-9962-9 |

ISSN: | 0178-4617 |

Projects: | Future Ocean |

Date Deposited: | 16 Mar 2017 07:30 |

Last Modified: | 22 May 2019 12:33 |

URI: | http://oceanrep.geomar.de/id/eprint/36073 |

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