Approximation algorithm for the multicovering problem

Abstract

Let $\mathcal {H}=(V,\mathcal {E})$ be a hypergraph with maximum edge size $\ell$ and maximum degree $\varDelta$. For a given positive integers $b_v$, $v\in V$, a set multicover in $\mathcal {H}$ is a set of edges $C \subseteq \mathcal {E}$ 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 $\varDelta$ and $b:=\min _{v\in V}b_{v}$, the problem of set multicover is not approximable within a ratio less than $\delta :=\varDelta -b+1$, unless $\mathcal {P}=\mathcal {NP}$. 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 $\max \left\{ \frac{148}{149}\delta , \left( 1- \frac{ (b-1)e^{\frac{\delta }{4}}}{94\ell } \right) \delta \right\}$ for $b\ge 2$ and $\delta \ge 3$. Our result not only improves over the approximation ratio presented by El Ouali et al. (Algorithmica 74:574, 2016) but it’s more general since we set no restriction on the parameter $\ell$. Moreover we present a further polynomial time algorithm with an approximation ratio of $\frac{5}{6}\delta$ for hypergraphs with $\ell \le (1+\epsilon )\bar{\ell }$ for any fixed $\epsilon \in [0,\frac{1}{2}]$, where $\bar{\ell }$ 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.