Abstract
Minimum Dominating Set and Minimum Connected Dominating Set are classic graph problems that have been studied extensively in the literature. These two problems and their various variants are NP-hard in a general graph, and for some of them greedy approximation algorithms have been proposed. In this paper, by designing two potential functions that enjoy submodularity or a weak submodularity, we propose a unified O(lnδ)-approximation algorithm for a generalized Minimum (Connected) Dominating Set that includes Minimum (Connected) Dominating Set, Minimum (Connected) Total Dominating Set, Minimum (Connected) *-Dominating Set and Minimum (Connected) Positive Influence Dominating Set, where δ is the maximum node degree of the input graph. For each specific version of the generalized Minimum (Connected) Dominating Set, the unified algorithm either matches the best one of existing approximation algorithms, or gives the first approximation solution.
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