Approximability of capacitated network design

In the capacitated survivable network design problem (Cap-SNDP), we are given an undirected multi-graph where each edge has a capacity and a cost. The goal is to find a minimum cost subset of edges that satisfies a given set of pairwise minimum-cut requirements. Unlike its classical special case of SNDP when all capacities are unit, the approximability of Cap-SNDP is not well understood; even in very restricted settings no known algorithm achieves a o(m) approximation, where m is the number of edges in the graph. In this paper, we obtain several new results and insights into the approximability of Cap-SNDP. We give an O(logn) approximation for a special case of Cap-SNDP where the global minimum cut is required to be at least R, by rounding the natural cut-based LP relaxation strengthened with valid knapsack-cover inequalities. We then show that as we move away from global connectivity, the single pair case (that is, when only one pair (s,t) has positive connectivity requirement) captures much of the difficulty of Cap-SNDP: even strengthened with KC inequalities, the LP has an Ω(n) integrality gap. Furthermore, in directed graphs, we show that single pair Cap-SNDP is 2^{log 1-δ n}-hard to approximate for any fixed constant δ>0. We also consider a variant of the Cap-SNDP in which multiple copies of an edge can be bought: we give an O(logk) approximation for this case, where k is the number of vertex pairs with non-zero connectivity requirement. This improves upon the previously known O( min {k, logR _max })-approximation for this problem when the largest minimum-cut requirement, namely R_max , is large. On the other hand, we observe that the multiple copy version of Cap-SNDP is Ω(log log n)-hard to approximate even for the single-source version of the problem.