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.