Approximation algorithm for the multicovering problem

Let $$\mathcal {H}=(V,\mathcal {E})$$ be a hypergraph with maximum edge size $$\ell $$ and degree $$\varDelta . For given positive integers $$b_v$$ , $$v\in V$$ set multicover in {H}$$ is of edges $$C \subseteq \mathcal {E}$$ such that every vertex v V belongs to at least C. Set the problem finding minimum-cardinality multicover. Peleg, Schechtman Wool conjectured for any fixed $$b:=\min _{v\in V}b_{v}$$ not approximable within ratio less than $$\delta :=\varDelta -b+1$$ unless {P}=\mathcal {NP}$$ Hence it’s challenge explore which classes conjecture doesn’t hold. We present polynomial time algorithm combines deterministic threshold conditioned randomized rounding steps. Our yields an approximation $$\max \left\{ \frac{148}{149}\delta \left( 1- \frac{ (b-1)e^{\frac{\delta }{4}}}{94\ell } \right) \delta \right\} $$b\ge 2$$ \ge 3$$ result only improves over presented by El Ouali et al. (Algorithmica 74:574, 2016) but more general since we no restriction on parameter Moreover further $$\frac{5}{6}\delta hypergraphs \le (1+\epsilon )\bar{\ell }$$ $$\epsilon \in [0,\frac{1}{2}]$$ where $$\bar{\ell average size. The analysis this relies matching/covering duality due Ray-Chaudhuri (1960), convert into approximative form. second performance disprove Peleg large subclass hypergraphs.