Approximation algorithms for minimum-cost k-(S, T ) connected digraphs

We introduce a model for NP-hard problems pertaining to the connectivity of graphs, and design approximation algorithms for some of the key problems in this model. One of the well-known NP-hard problems is to find a minimum-cost strongly connected spanning subgraph of a directed network. In the minimum-cost k-(S, T ) connected digraph (abbreviated, k-(S, T ) connectivity) problem we are given a positive integer k, a directed graph G = (V,E) with non-negative costs on the edges, and two subsets S, T of V ; the goal is to find a subset of edges Ê of minimum cost such that the subgraph (V, Ê) has k edge-disjoint directed paths from each vertex in S to each vertex in T . When k = 1 and S = T = V , we get the minimum-cost strongly connected spanning subgraph problem. Our model of k-(S, T ) connectivity, in its full generality, is at least as hard for approximation as the label-cover problem, even when the connectivity parameter k is one. We give a simple reduction from the directed Steiner network problem, and it is well known that the latter problem is at least as hard as the label-cover problem. Rather than focusing on this general version of the problem, most of our results focus on a specialized version that we call the standard version, where every edge of positive cost has its tail in S and its head in T . This version of the problem captures NP-hard problems such as the minimum-cost k-vertex connected spanning subgraph problem. We call it the standard version because a still further specialization of it was introduced and studied by Frank and Jordan more than fifteen years ago; in their model, which is polynomial-time solvable, every edge from S to T is present and has unit cost. One of our key contributions is an approximation algorithm with a guarantee ofO((log k)(log n)) for the standard version of the k-(S, T ) connectivity problem. For k = 1, we give a simple 2-approximation algorithm that generalizes a well-known 2-approximation algorithm for the minimum-cost strongly connected spanning subgraph problem. For k = 2, we give a simple 4-approximation algorithm, though the analysis is nontrivial. Besides the standard version, we study another version that is intermediate between the standard version and the problem in its full generality (which is label-cover hard). In the relaxed version of the (S, T ) connectivity problem, each edge of positive cost has its head in T but there is no restriction on the tail. We study the relaxed version with the connectivity parameter k fixed at one, and observe that this version is at least as hard to approximate as the directed Steiner tree problem. We match this by giving an algorithm that achieves an approximation guarantee of α(n) + 1 for the relaxed (S, T ) connectivity problem, where α(n) denotes the best approximation guarantee available for the directed Steiner tree problem. The algorithm is simple, but not the analysis. The key to the analysis is a structural result that ∗Dept. of Comb. & Opt., U. Waterloo, Waterloo ON Canada N2L 3G1. Email: [email protected] †Dept. of Computer Science, McGill University, Montreal QC Canada Email: [email protected]