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.

令\（\ mathcal {H} =（V，\ mathcal {E}）\）是最大边尺寸为\（\ ell \）和最大度数为\（\ varDelta \）的超图。对于给定的正整数\（b_v \），\（v \ in V \），\（\ mathcal {H} \）中的set multicover是边的集合\（C \ subseteq \ mathcal {E} \）这样，每一个顶点v在v属于至少\（b_v \）边缘在ç。集合多重覆盖是找到最小基数集合多重覆盖的问题。Peleg，Schechtman和Wool猜想对于任何固定的\（\ varDelta \）和\（b：= \ min _ {v \ in V} b_ {v} \），在小于\（\ delta：= \ varDelta -b + 1 \）的比率内，设置multicover的问题不是近似的，除非\（\ mathcal {P} = \ mathcal {NP} \）。因此，探索该猜想不适合哪些类的超图是一个挑战。我们提出了一种针对集合多重覆盖问题的多项式时间算法，该算法结合了确定性阈值算法和条件随机舍入步骤。我们的算法得出\（\ max \ left \ {\ frac {148} {149} \ delta，\ left（1- \ frac {（b-1）e ^ {\ frac {\ delta} {4 }}} {94 \ ELL} \右）\增量\右\} \）为\（b \ GE 2 \）和\（\增量\ GE 3 \）。我们的结果不仅改善了El Ouali等人提出的近似比率。（Algorithmica 74：574，2016），但由于我们对参数\（\ ell \）没有设置任何限制，因此它更具通用性。此外，对于具有\（\ ell \ le（1+ \ epsilon）\ bar {\ ell} \）的超图，我们提出了一种进一步的多项式时间算法，其近似比为\（\ frac {5} {6} \ delta \）对于任何固定的\（\ epsilon \ in [0，\ frac {1} {2}] \），其中\（\ bar {\ ell} \）是平均边缘尺寸。该算法的分析依赖于归因于Ray-Chaudhuri（1960）的匹配/覆盖对偶，我们将其转换为近似形式。第二种表现反驳了Peleg等人的猜想。用于超图的大子类。