| Summary: | 博士 === 國立臺灣大學 === 資訊工程學研究所 === 107 === In this dissertation, we aim to address the approximability of the capacitated vertex cover (CVC) problem, i.e., a generic demand-to-service assignment model of the classical vertex cover problem, when certain constraints of interest are relaxed. In CVC, we are given a hypergraph H = (V, E) with a maximum edge size f. Each (hyper)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, or cover, such that the demands of the edges can be met by the capacities of the vertices and the multiplicity of each vertex does not exceed its available multiplicity. Depending on whether the available multiplicity of each vertex is limited, the work on CVC falls mainly in two categories: (1) soft capacity (SCVC), where the multiplicity of each vertex is unlimited, and (2) hard capacity (HCVC), where the available multiplicity of each vertex is limited.
We give an overview of the proposed models and methods for CVC, and summarize related research works for SCVC, HCVC and other variations in recent years. And our main results are focused on different relaxed constraint models of HCVC. When each vertex is unweighted, we have a (δ + 2)-approximation algorithm, where δ is the maximum vertex degree. For augmenting the available multiplicity by a factor of k ≥ 2, a cover with a cost ratio of (1 + 1/(k-1))(f − 1) to
the optimal cover for the original instance can be obtained. For partial cover, as the demand served is at least the ratio of (1 − ϵ), we have an O(1/ϵ)f-approximation algorithm for HCVC.
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