For a given $\mathbf{b}\in {\mathbb{N}}^n$ and a hypergraph $\mathcal{H}=(V,\mathcal{E})$ with maximum degree Δ, the b-set multicover problem which we are concerned with in this paper may be stated as follows: find a minimum cardinality subset $\mathcal{C}\subseteq \mathcal{E}$ such that no vertex $v_i\in V$ is contained in less than $b_i$ hyperedges of $\mathcal{C}$. Peleg, Schechtman, and Wool (1997) conjectured that for any fixed Δ and $b={\min }_i{b}_i$, the problem cannot be approximated with a ratio strictly smaller than $\delta :=\mathrm{\Delta }-b+1$, unless $\mathcal{P}=\mathcal{NP}$. In this paper, we show that the conjecture of Peleg et al. is not true on ℓ-uniform hypergraphs by presenting a polynomial-time approximation algorithm based on a matching/covering duality for hypergraphs due to Ray-Chaudhuri (1960), which we convert into an approximative form. The given algorithm yields a ratio smaller than $(1-\frac{b}{(\ell +1)\mathrm{\Delta }})\frac{\mathrm{\Delta }}{b}$. Moreover, we prove that the lower bound conjectured by Peleg et al. holds for regular hypergraphs under the unique games conjecture.

对于给定 $\mathbf{b}\in {\mathbb{ñ}}^{ñ}$ 和一个超图 $\mathcal{H}=（V，\mathcal{Ë}）$在最大度为Δ的情况下，我们在本文中关注的b-集多重覆盖问题可以表示为：找到最小基数子集$\mathcal{C}\subseteq \mathcal{Ë}$ 这样没有顶点 $v_{\mathrm{一世}}\in V$ 包含在少于 $b_{\mathrm{一世}}$ 的超边缘 $\mathcal{C}$。Peleg，Schechtman和Wool（1997）推测，对于任何固定的Δ和$b={\mathrm{分}}_{\mathrm{一世}}{b}_{\mathrm{一世}}$，问题的比率不能严格小于 $\delta ：=\mathrm{\Delta }-b+\mathrm{1个}$，除非 $\mathcal{P}=\mathcal{NP}$。在本文中，我们证明了Peleg等人的猜想。不在真ℓ通过呈现基于用于由于雷-乔赫里（1960），我们转换成近似形式超图的匹配/覆盖二元多项式时间近似算法-uniform超图。给定的算法产生的比率小于$（\mathrm{1个}-\frac{b}{（\ell +\mathrm{1个}）\mathrm{\Delta }}）\frac{\mathrm{\Delta }}{b}$。此外，我们证明了Peleg等人的下界。在独特的游戏猜想下持有常规的超图。