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 tackles the Vertex‑Connectivity Survivable Network Design Problem (VCSNDP), where one is given an undirected graph G = (V,E), a non‑negative integer requirement r(u,v) for every unordered pair of vertices, and a cost c(e) on each edge. The objective is to select a minimum‑cost edge subset H⊆E such that, in the subgraph (V,H), every pair (u,v) is connected by r(u,v) internally vertex‑disjoint paths. The maximum requirement over all pairs is denoted by k.

Previously, Chuzhoy and Khanna introduced a randomized algorithm achieving an O(k³·log n) approximation. Their method relies on two main components: (i) a random sampling of “routing trees” that can carry the required vertex‑disjoint flows, and (ii) a rounding step that picks a cheapest sub‑tree from each sampled routing tree. The analysis shows that, in expectation, the total cost does not exceed O(k³·log n) times the optimum. However, the reliance on randomness makes the algorithm unsuitable for environments that demand reproducibility or deterministic guarantees.

The present work derandomizes the entire framework while preserving the same approximation factor. The authors introduce three technical innovations.

1. ε‑Network Flow Transformation – Each requirement r(u,v) is expressed as an integer flow in a vertex‑capacity network where every vertex has capacity k. By allowing a small additive error ε, the flow can be decomposed into a polynomial‑size collection of “basic routing trees”. This conversion is deterministic and can be performed in time polynomial in |V|, |E|, k, and log n.

2. Deterministic Generation of Routing‑Tree Candidates – Instead of sampling trees at random, the algorithm enumerates all candidate trees and assigns each a “efficiency score” η(T) = Δr(T)/c(T), where Δr(T) is the amount of unsatisfied demand that would be satisfied by adding tree T, and c(T) is its total edge cost. Trees are then processed in decreasing order of η(T). Computing η(T) for all candidates can be done using standard shortest‑path and matroid‑intersection subroutines, staying within the same polynomial bound as the randomized version.

3. Round‑Robin Deterministic Selection – The algorithm iteratively adds the highest‑efficiency tree that still contributes to unmet demand, then updates the residual requirements and recomputes efficiencies. A round‑robin schedule guarantees that no single tree dominates the solution, which is crucial for the analysis. By proving that each iteration reduces the remaining “deficit” by at least a 1/k fraction of the optimal cost, the authors show that after O(k·log n) iterations the residual demand is zero. Consequently the total cost is bounded by O(k³·log n)·OPT, matching the randomized bound.

For the single‑source variant (all non‑zero requirements involve a distinguished source s), the authors observe that every feasible routing tree must be s‑rooted. This restriction shrinks the candidate set dramatically, allowing a tighter analysis. The same deterministic selection procedure now yields an O(k²·log n) approximation, improving on the general‑case bound by a factor of k.

The paper also provides a thorough complexity analysis. The deterministic algorithm runs in O(m·poly(k,log n)) time and uses O(m + n·k) space, identical to the randomized counterpart. Empirical evaluation on synthetic random graphs and real‑world network topologies confirms that the deterministic method incurs only a negligible overhead (≈5 % higher cost on average) while delivering reproducible solutions. In the single‑source experiments, the O(k²·log n) algorithm consistently outperforms the O(k³·log n) version by roughly 30 % in total cost.

In summary, the authors achieve three major contributions: (1) a fully deterministic O(k³·log n) approximation for the general VCSNDP, (2) a deterministic O(k²·log n) approximation for the single‑source case, and (3) a clean, modular framework that replaces random sampling with an ε‑network‑flow‑based decomposition and an efficiency‑driven round‑robin selection. This work eliminates the need for randomness in a classic survivable network design problem, paving the way for practical, provably‑good implementations in mission‑critical networking and infrastructure planning. Future directions include extending the technique to directed graphs, hypergraphs, or settings where edge‑capacities (instead of vertex‑capacities) dominate the connectivity constraints.