Randomized Approximation for the Set Multicover Problem in Hypergraphs

Let b ∈ N ≥ 1 and let H = ( V , E ) be a hypergraph with maximum vertex degree Δ and maximum edge size l . A set b -multicover in H is a set of edges C ⊆ E such that every vertex in V belongs to at least b edges in C . S E T b - M U L T I C O V E R is the problem of finding a set b -multicover of mi...

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Veröffentlicht in:Algorithmica 2016-02, Vol.74 (2), p.574-588
Hauptverfasser: El Ouali, Mourad, Munstermann, Peter, Srivastav, Anand
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Sprache:eng
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Zusammenfassung:Let b ∈ N ≥ 1 and let H = ( V , E ) be a hypergraph with maximum vertex degree Δ and maximum edge size l . A set b -multicover in H is a set of edges C ⊆ E such that every vertex in V belongs to at least b edges in C . S E T b - M U L T I C O V E R 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 δ · ( 1 - ( c n ) 1 δ ) , where δ : = Δ - b + 1 and c > 0 is a constant. As this ratio depends on the instance size n and tends to δ as n tends to ∞ , it remained an open problem whether an approximation ratio of δ α with a constant α < 1 can be proved. In fact, the authors conjectured that for any fixed Δ and b , the problem is not approximable within a ratio smaller than δ , unless P = NP . We present a randomized algorithm of hybrid type for S E T b - M U L T I C O V E R , b ≥ 2 , combining LP-based randomized rounding with greedy repairing, and achieve an approximation ratio of δ · 1 - 11 ( Δ - b ) 72 l for hypergraphs with maximum edge size l ∈ O max { ( n b ) 1 5 , n 1 4 } . In particular, for all hypergraphs where l is constant , we get an α δ -ratio with constant α < 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 δ .
ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-014-9962-9