A Deterministic Algorithm for the Vertex Connectivity Survivable Network Design Problem

In the vertex connectivity survivable network design problem we are given an undirected graph G = (V,E) and connectivity requirement r(u,v) for each pair of vertices u,v. We are also given a cost function on the set of edges. Our goal is to find the minimum cost subset of edges such that for every pair (u,v) of vertices we have r(u,v) vertex disjoint paths in the graph induced by the chosen edges. Recently, Chuzhoy and Khanna presented a randomized algorithm that achieves a factor of O(k^3 log n) for this problem where k is the maximum connectivity requirement. In this paper we derandomize their algorithm to get a deterministic O(k^3 log n) factor algorithm. Another problem of interest is the single source version of the problem, where there is a special vertex s and all non-zero connectivity requirements must involve s. We also give a deterministic O(k^2 log n) algorithm for this problem.

💡 Research Summary

The paper addresses the Vertex‑Connectivity Survivable Network Design Problem (VC‑SNDP), where one is given an undirected graph G = (V,E), a cost on each edge, and for every unordered pair of vertices (u,v) a connectivity requirement r(u,v) ≤ k. The objective is to select a minimum‑cost edge set such that for each pair (u,v) the induced subgraph contains r(u,v) vertex‑disjoint paths.

Previously, Chuzhoy and Khanna (FOCS 2011) presented a randomized O(k³·log n)‑approximation algorithm. Their method relies on constructing a “good family” of terminal subsets that satisfies a combinatorial property (weak goodness). Verifying that a given family is good is NP‑hard, which prevents a straightforward derandomization.

The authors introduce a new structural notion called strong goodness. A bipartite graph G = (L ∪ R,E) satisfies (α,β)‑strong goodness if (i) every two distinct left vertices have at least α common neighbors, and (ii) every three distinct left vertices have at most β common neighbors. If α/β > k, strong goodness implies weak goodness, so any graph with this property yields a valid good family.

To build such a graph, each left vertex ℓ_i is assigned a string w_i of length γ over an alphabet A of size |A| = c·k (c is a sufficiently large constant). Right vertices correspond to pairs (j,c) where j∈