On finding highly connected spanning subgraphs

TL;DR

This paper studies the survivable network design problem focusing on finding minimal weight, highly connected spanning subgraphs with fewer edges, and provides new fixed-parameter algorithms and structural insights.

Contribution

It introduces the first fixed-parameter tractable algorithm and polynomial kernel for a connectivity problem related to SNDP, with significant structural results.

Findings

Algorithm for mbda-ECS runs in 2^{O(k \u2217 iglog k)}  |V(G)|^{O(1)}

First polynomial compression for mbda-ECS on unweighted graphs

Fixed-parameter tractable algorithm and polynomial kernel for a variant of the Minimum Equivalent Graph problem

Abstract

In the Survivable Network Design Problem (SNDP), the input is an edge-weighted (di)graph $G$ and an integer $r_{uv}$ for every pair of vertices $u,v\in V(G)$. The objective is to construct a subgraph $H$ of minimum weight which contains $r_{uv}$ edge-disjoint (or node-disjoint) $u$-$v$ paths. This is a fundamental problem in combinatorial optimization that captures numerous well-studied problems in graph theory and graph algorithms. In this paper, we consider the version of the problem where we are given a $\lambda$-edge connected (di)graph $G$ with a non-negative weight function $w$ on the edges and an integer $k$, and the objective is to find a minimum weight spanning subgraph $H$ that is also $\lambda$-edge connected, and has at least $k$ fewer edges than $G$. In other words, we are asked to compute a maximum weight subset of edges, of cardinality up to $k$, which may be safely deleted from $G$. Motivated by this question, we investigate the connectivity properties of $\lambda$-edge connected (di)graphs and obtain algorithmically significant structural results. We demonstrate the importance of our structural results by presenting an algorithm running in time $2^{O(k \log k)} |V(G)|^{O(1)}$ for $\lambda$-ECS, thus proving its fixed-parameter tractability. We follow up on this result and obtain the {\em first polynomial compression} for $\lambda$-ECS on unweighted graphs. As a consequence, we also obtain the first fixed parameter tractable algorithm, and a polynomial kernel for a parameterized version of the classic Mininum Equivalent Graph problem. We believe that our structural results are of independent interest and will play a crucial role in the design of algorithms for connectivity-constrained problems in general and the SNDP problem in particular.