On k-connectivity problems with sharpened triangle inequality

The k-connectivity problem is to find a minimum-cost k-edge- or k-vertex-connected spanning subgraph of an edge-weighted, undirected graph G for any given G and k. Here, we consider its NP-hard subproblems with respect to the parameter @b, with 12<@b<1, where G=(V,E) is a complete graph with a cost function c satisfying the sharpened triangle inequality c({u,v})=<@[email protected]?(c({u,w})+c({w,v})) for all u,v,[email protected]?V. First, we give a simple linear-time approximation algorithm for these optimization problems with approximation ratio @[email protected] for any 12=<@b<1, which improves the known approximation ratios for 12<@b<23. The analysis of the algorithm above is based on a rough combinatorial argumentation. As the main result of this paper, for k=3, we sophisticate the combinatorial consideration in order to design a (1+5([email protected])9([email protected])+O(1|V|))-approximation algorithm for the 3-connectivity problem on graphs satisfying the sharpened triangle inequality for 12=<@b=<23. As part of the proof, we show that for each spanning 3-edge-connected subgraph H, there exists a spanning 3-regular 2-vertex-connected subgraph H^' of at most the same cost, and H can be transformed into H^' efficiently.

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