Approximability of Capacitated Network Design

In the {\em 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.

💡 Research Summary

The paper investigates the approximability of the Capacitated Survivable Network Design Problem (Cap‑SNDP), a natural generalization of the classical Survivable Network Design Problem (SNDP) where each edge carries an arbitrary integer capacity in addition to a cost. The goal is to select a minimum‑cost subgraph that satisfies, for every unordered pair of vertices (i, j), a prescribed minimum‑cut capacity requirement R_{ij}. While SNDP (unit capacities) admits a celebrated 2‑approximation via Jain’s iterative rounding, the presence of heterogeneous capacities dramatically changes the landscape: even the most restricted special cases lack sub‑linear approximations.

The authors first focus on a uniform‑requirement variant, called the Cap‑R‑Connected Subgraph problem, where every pair must have a minimum cut of at least R. They formulate the standard cut‑based linear program (LP) and observe that it can have an unbounded integrality gap, even on a two‑vertex parallel‑edge instance. To overcome this, they strengthen the LP with knapsack‑cover (KC) inequalities, which enforce that any collection of edges crossing a cut must collectively provide enough capacity to meet the requirement. Although separating all KC constraints is NP‑hard, they show that a violated inequality can be identified whenever the fractional solution fails certain “useful” properties. The rounding algorithm leverages Karger’s theorem on the number of small cuts in undirected graphs: by randomly sampling a modest number of cuts (O(log n) in expectation) they guarantee that every cut of interest is represented. A randomized rounding scheme then selects edges to satisfy all sampled cuts, incurring at most an O(log n) factor over the LP optimum. Consequently, they obtain a randomized O(log n)‑approximation for the uniform‑requirement case, and an O(γ log n)‑approximation when requirements lie in the interval