An $O(k^{3} log n)$-Approximation Algorithm for Vertex-Connectivity Survivable Network Design

In the Survivable Network Design problem (SNDP), we are given an undirected graph $G(V,E)$ with costs on edges, along with a connectivity requirement $r(u,v)$ for each pair $u,v$ of vertices. The goal is to find a minimum-cost subset $E^*$ of edges, that satisfies the given set of pairwise connectivity requirements. In the edge-connectivity version we need to ensure that there are $r(u,v)$ edge-disjoint paths for every pair $u, v$ of vertices, while in the vertex-connectivity version the paths are required to be vertex-disjoint. The edge-connectivity version of SNDP is known to have a 2-approximation. However, no non-trivial approximation algorithm has been known so far for the vertex version of SNDP, except for special cases of the problem. We present an extremely simple algorithm to achieve an $O(k^3 \log n)$-approximation for this problem, where $k$ denotes the maximum connectivity requirement, and $n$ denotes the number of vertices. We also give a simple proof of the recently discovered $O(k^2 \log n)$-approximation result for the single-source version of vertex-connectivity SNDP. We note that in both cases, our analysis in fact yields slightly better guarantees in that the $\log n$ term in the approximation guarantee can be replaced with a $\log \tau$ term where $\tau$ denotes the number of distinct vertices that participate in one or more pairs with a positive connectivity requirement.

💡 Research Summary

The paper addresses the Vertex‑Connectivity Survivable Network Design Problem (VC‑SNDP), where we are given an undirected graph with edge costs and a connectivity requirement r(u,v)≤k for each pair of vertices. The goal is to find a minimum‑cost edge set that guarantees r(u,v) vertex‑disjoint paths for every required pair. While the edge‑connectivity version (EC‑SNDP) enjoys a classic 2‑approximation via Jain’s iterative rounding, VC‑SNDP has resisted non‑trivial approximations beyond a few special cases.

The authors present a remarkably simple randomized algorithm that achieves an O(k³ log n) approximation for the general VC‑SNDP, where k is the maximum demand and n the number of vertices. The key idea is to reduce the vertex‑connectivity instance to a collection of element‑connectivity instances, each of which can be solved by the known 2‑approximation algorithm for element‑connectivity SNDP (Fleischer et al., 2007). By solving all the reduced instances and taking the union of their edge sets, the algorithm obtains a feasible solution whose cost is at most 2p·OPT, where p = Θ(k³ log n).

Construction details: create p copies G₁,…,G_p of the original graph. For each copy define a terminal subset T_i ⊆ T (the set of vertices with non‑zero demand). Each terminal t independently selects q = p/(2k) = Θ(k² log n) indices uniformly at random (with repetitions) and belongs to all T_i whose index appears in its selection. This random assignment yields a family {T_i} that, with high probability, satisfies the “good family” property: for any source‑sink pair (s,t) and any set X of at most k‑1 vertices, there exists an i such that s,t ∈ T_i and X∩T_i = ∅. In other words, at least one copy isolates the pair from any potential small cut.

For each copy G_i, the induced element‑connectivity instance has the same demand values for pairs inside T_i and zero elsewhere. Applying the 2‑approximation algorithm yields a set E_i of edges that makes every required pair element‑connected within G_i. Because X consists only of non‑terminal vertices for that instance, the element‑connectivity guarantee ensures that removal of X does not disconnect s from t. Taking E* = ⋃_{i=1}^p E_i therefore satisfies all vertex‑connectivity requirements of the original problem.

The probabilistic analysis uses two types of “bad events”. Event E₁(s,t,X) occurs when the random index set of s overlaps too much with that of X; Chernoff bounds give Pr