Fast Exact Algorithms for Survivable Network Design with Uniform Requirements

Abstract

We design exact algorithms for the following two problems in survivable network design: (i) designing a minimum cost network with a desired value of edge connectivity, which is called Minimum Weight $\lambda$-connected Spanning Subgraph and (ii) augmenting a given network to a desired value of edge connectivity at a minimum cost which is called Minimum Weight $\lambda$-connectivity Augmentation. Many well known problems such as Minimum Spanning Tree, Hamiltonian Cycle, Minimum 2-Edge Connected Spanning Subgraph and Minimum Equivalent Digraph reduce to these problems in polynomial time. It is easy to see that a minimum solution to these problems contains at most $2 \lambda (n-1)$ edges. Using this fact one can design a brute-force algorithm which runs in time $2^{\mathcal {O}(\lambda n(\log n + \log \lambda )}$. However no better algorithms were known. In this paper, we give the first single exponential time algorithm for these problems, i.e. running in time $2^{\mathcal {O}(\lambda n)}$, for both undirected and directed networks. Our results are obtained via well known characterizations of $\lambda$-connected graphs, their connections to linear matroids and the recently developed technique of dynamic programming with representative sets.

Supported by “Parameterized Approximation” ERC Starting Grant 306992 and “Rigorous Theory of Preprocessing” ERC Advanced Investigator Grant 267959.