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, 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 + ?], for an arbitrary small ? > 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 s-triangle inequality, then a (2/3 + 1/3, s/1-s)-approximation algorithm is designed. This ratio tends to 1 with s tending to 1/2, and it improves the previous known bound of 3/2, holding for graphs satisfying the triangle inequality, as soon as s < 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 s-triangle inequality with s < 2/3, then this ratio can be improved to s/1-s.