Approximation algorithm for the Multicovering Problem
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Let H=(V,E) be a hypergraph with maximum edge size ℓ and maximum degree Δ. For given numbers b_v∈N_≥ 2, v∈ V, a set multicover in H is a set of edges C ⊆E such that every vertex v in V belongs to at least b_v edges in C. Set Multicover is the problem of finding a minimum-cardinality set multicover. Peleg, Schechtman and Wool conjectured that for any fixed Δ and b:=min_v∈ Vb_v, the problem of is not approximable within a ratio less than δ:=Δ-b+1, unless P =NP. Hence it's a challenge to explore for which classes of hypergraph the conjecture doesn't hold. We present a polynomial time algorithm for the Set Multicover problem which combines a deterministic threshold algorithm with conditioned randomized rounding steps. Our algorithm yields an approximation ratio of max{148/149δ, (1- (b-1)e^δ/4/94ℓ)δ}. Our result not only improves over the approximation ratio presented by Srivastav et al (Algorithmica 2016) but it's more general since we set no restriction on the parameter ℓ. Moreover we present a further polynomial time algorithm with an approximation ratio of 5/6δ for hypergraphs with ℓ≤ (1+ϵ)ℓ̅ for any fixed ϵ∈ [0,1/2], where ℓ̅ is the average edge size. The analysis of this algorithm relies on matching/covering duality due to Ray-Chaudhuri (1960), which we convert into an approximative form. The second performance disprove the conjecture of peleg et al for a large subclass of hypergraphs.
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