On the hardness of constructing minimal 2-connected spanning subgraphs in complete graphs with sharpened triangle inequality

Abstract

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, for both the edge and 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 version (i.e., for arbitrary graphs) of the problem. 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 .