Repeated randomized algorithm for the Multicovering Problem

Let $\mathcal{H}=(V,\mathcal{E})$ be a hypergraph with maximum edge size $\ell$ and maximum degree $\Delta$. For given numbers $b_v\in \mathbb{N}_{\geq 2}$, $v\in V$, 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 unless $\cal{P} =\cal{NP}$, for any fixed $\Delta$ and $b:=\min_{v\in V}b_{v}$, no polynomial-time approximation algorithm for the Set multicover problem has an approximation ratio less than $\delta:=\Delta-b+1$. Hence, it's a challenge to know whether the problem of set multicover is not approximable within a ratio of $\beta \delta$ with a constant $\beta<1$. This paper proposes a repeated randomized algorithm for the Set multicover problem combined with an initial deterministic threshold step. Boosting success by repeated trials, our algorithm yields an approximation ratio of $ \max\left\{ \frac{15}{16}\delta, \left(1- \frac{(b-1)\exp\left(\frac{ 3\delta+1}{8}\right)}{72 \ell} \right)\delta\right\}$. The crucial fact is not only that our result improves over the approximation ratio presented by Srivastav et al (Algorithmica 2016) for any $\delta\geq 13$, but it's more general since we set no restriction on the parameter $\ell$. Furthermore, we prove that it is NP-hard to approximate the Set multicover problem on $\Delta$-regular hypergraphs within a factor of $(\delta-1-\epsilon)$. Moreover we show that the integrality gap for the Set multicover problem is at least $\frac{\ln_{2}(n+1)}{2b}$, which for constant $b$ is $\Omega(\ln n )$.