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. (Note that this problem generalizes the min-cost ¿-edge-connected subgraph problem, which is the special case of our problem when all capacities are unit.) Our result is based on a rounding of a natural cut-based LP relaxation strengthened with knapsack-cover (KC) inequalities. Our technique extends to give a similar approximation for a new network design problem that captures global minimum cut as a special case. We then show that as we move away from global connectivity, even for the single pair case (that is, when only one pair (s,t) has positive connectivity requirement), this strengthened LP has Ω(n) integrality gap.

We also consider a variant of 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,logRmax})-approximation when Rmax is large; here Rmax is the largest requirement. On the other hand, we observe that the multiple copy version of Cap-SNDP is Ω(loglogn)-hard to approximate even for the single-source version of the problem.