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# A Parameterized Algorithm for Vertex Connectivity Survivable Network Design Problem with Uniform Demands

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## Cite As

Jørgen Bang-Jensen, Kristine Vitting Klinkby, Pranabendu Misra, and Saket Saurabh. A Parameterized Algorithm for Vertex Connectivity Survivable Network Design Problem with Uniform Demands. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ESA.2023.13

## Abstract

In the Vertex Connectivity Survivable Network Design (VC-SNDP) problem, the input is a graph G and a function d: V(G) × V(G) → ℕ that encodes the vertex-connectivity demands between pairs of vertices. The objective is to find the smallest subgraph H of G that satisfies all these demands. It is a well-studied NP-complete problem that generalizes several network design problems. We consider the case of uniform demands, where for every vertex pair (u,v) the connectivity demand d(u,v) is a fixed integer κ. It is an important problem with wide applications. We study this problem in the realm of Parameterized Complexity. In this setting, in addition to G and d we are given an integer 𝓁 as the parameter and the objective is to determine if we can remove at least 𝓁 edges from G without violating any connectivity constraints. This was posed as an open problem by Bang-Jansen et.al. [SODA 2018], who studied the edge-connectivity variant of the problem under the same settings. Using a powerful classification result of Lokshtanov et al. [ICALP 2018], Gutin et al. [JCSS 2019] recently showed that this problem admits a (non-uniform) FPT algorithm where the running time was unspecified. Further they also gave an (uniform) FPT algorithm for the case of κ = 2. In this paper we present a (uniform) FPT algorithm any κ that runs in time 2^{O(κ² 𝓁⁴ log 𝓁)}⋅ |V(G)|^O(1). Our algorithm is built upon new insights on vertex connectivity in graphs. Our main conceptual contribution is a novel graph decomposition called the Wheel decomposition. Informally, it is a partition of the edge set of a graph G, E(G) = X₁ ∪ X₂ … ∪ X_r, with the parts arranged in a cyclic order, such that each vertex v ∈ V(G) either has edges in at most two consecutive parts, or has edges in every part of this partition. The first kind of vertices can be thought of as the rim of the wheel, while the second kind form the hub. Additionally, the vertex cuts induced by these edge-sets in G have highly symmetric properties. Our main technical result, informally speaking, establishes that "nearly edge-minimal’’ κ-vertex connected graphs admit a wheel decomposition - a fact that can be exploited for designing algorithms. We believe that this decomposition is of independent interest and it could be a useful tool in resolving other open problems.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
##### Keywords
• Parameterized Complexity
• Vertex Connectivity
• Network Design

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## References

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