On k-connectivity problems with sharpened triangle inequality

Abstract

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 β, with 1 2 < β < 1 , where G = ( V , E ) is a complete graph with a cost function c satisfying the sharpened triangle inequality c ( { u , v } ) ⩽ β ⋅ ( c ( { u , w } ) + c ( { w , v } ) ) for all u , v , w ∈ V . First, we give a simple linear-time approximation algorithm for these optimization problems with approximation ratio β 1 − β for any 1 2 ⩽ β < 1 , which improves the known approximation ratios for 1 2 < β < 2 3 . 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 ( 2 β − 1 ) 9 ( 1 − β ) + O ( 1 | V | ) ) -approximation algorithm for the 3-connectivity problem on graphs satisfying the sharpened triangle inequality for 1 2 ⩽ β ⩽ 2 3 . 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.