具有硬容量的容量约束覆盖的O（f）双逼近
O(f) Bi-Approximation for Capacitated Covering with Hard Capacities
摘要：
本文研究超图上具有硬容量约束的容量约束顶点覆盖问题。在这个问题中，我们给出了一个超图$G=(V,E)$，它具有最大边长$f$。每个边与需求相关联，每个顶点与权重（成本）、容量和可用多样性相关联。我们的目标是找到一个最小权的顶点集，使边的需求可以覆盖的顶点的能力和每个顶点的重数不超过其可用的重数。在本文中，我们提出了一个$O(f)$双近似的VC-HC，给出了一个折衷的增广多重数和成本的覆盖。特别是，我们表明，通过增加可用的多重性的一个因素$k \ge 2$，一个覆盖成本比为$\Big(1+\frac{1}{k-1}\Big)(f-1)$的最佳覆盖的原始实例可以得到。这改进了先前的结果，其具有通过将可用多重性增大因子$f$而得到的成本比$f^2$。
英文摘要：
We consider capacitated vertex cover with hard capacity constraints (VC-HC) on hypergraphs. In this problem we are given a hypergraph $G=(V,E)$ with a maximum edge size $f$. Each edge is associated with a demand and each vertex is associated with a weight (cost), a capacity, and an available multiplicity. The objective is to find a minimum-weight vertex multiset such that the demands of the edges can be covered by the capacities of the vertices and the multiplicity of each vertex does not exceed its available multiplicity. In this paper we present an $O(f)$ bi-approximation for VC-HC that gives a trade-off on the number of augmented multiplicity and the cost of the resulting cover. In particular, we show that, by augmenting the available multiplicity by a factor of $k \ge 2$, a~cover with a cost ratio of $\Big(1+\frac{1}{k-1}\Big)(f-1)$ to the optimal cover for the original instance can be obtained. This improves over a previous result, which has a cost ratio of $f^2$ via augmenting the available multiplicity by a factor of $f$.
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