On the Hardness of Constructing Minimal 2-Connected Spanning Subgraphs in Complete Graphs with Sharpened Triangle Inequality
In this paper we investigate the problem of finding a 2- connected spanning subgraph of minimal cost in a complete and weighted graph G. This problem is known to be APX-hard, both for the edge- and for the vertex-connectivity case. Here we prove that the APX-hardness still holds even if one restricts the edge costs to an interval [1,1 + ε], for an arbitrary small ε > 0. This result implies the first explicit lower bound on the approximability of the general problems.
On the other hand, if the input graph satisfies the sharpened β-triangle inequality, then a (2/3 + 1/3 . β/1-β)-approximation algorithm is designed. This ratio tends to 1 with β tending to 1/2, and it improves the previous known bound of 3/2, holding for graphs satisfying the triangle inequality, as soon as β < 5/7.
Furthermore, a generalized problem of increasing to 2 the edge-connectivity of any spanning subgraph of G by means of a set of edges of minimum cost is considered. This problem is known to admit a 2-approximation algorithm. Here we show that whenever the input graph satisfies the sharpened β-triangle inequality with β < 2/3, then this ratio can be improved to β/1-β.
KeywordsApproximation algorithm inapproximability minimum-cost biconnected spanning subgraph augmentation
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