k-Edge-Connectivity: Approximation and LP Relaxation

In the k-edge-connected spanning subgraph problem we are given a graph (V, E) and costs for each edge, and want to find a minimum-cost subset F of E such that (V, F) is k-edge-connected. We show there is a constant eps > 0 so that for all k > 1, finding a (1 + eps)-approximation for k-ECSS is NP-hard, establishing a gap between the unit-cost and general-cost versions. Next, we consider the multi-subgraph cousin of k-ECSS, in which we purchase a multi-subset F of E, with unlimited parallel copies available at the same cost as the original edge. We conjecture that a (1 + Theta(1/k))-approximation algorithm exists, and we describe an approach based on graph decompositions applied to its natural linear programming (LP) relaxation. The LP is essentially equivalent to the Held-Karp LP for TSP and the undirected LP for Steiner tree. We give a family of extreme points for the LP which are more complex than those previously known.