Reoptimization of Weighted Graph and Covering Problems

Given an instance of an optimization problem and a good solution of that instance, the reoptimization is a concept of analyzing how does the solution change when the instance is locally modified. We investigate reoptimization of the following problems: Maximum Weighted Independent Set, Maximum Weighted Clique, Minimum Weighted Dominating Set, Minimum Weighted Set Cover and Minimum Weighted Vertex Cover. The local modifications we consider are addition or removal of a constant number of edges to the graph, or elements to the covering sets in case of Set Cover problem. We present the following results: 1 We provide a PTAS for reoptimization of the unweighted versions of the aforementioned problems when the input solution is optimal. 1 We provide two general techniques for analyzing approximation ratio of the weighted reoptimization problems. 1 We apply our techniques to reoptimization of the considered optimization problems and obtain tight approximation ratios in all the cases.

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