Approximation algorithms and an integer program for multi-level graph spanners

Abstract

Given a weighted graph $G(V,E)$ and $t\ge 1$, a subgraph $H$ is a \emph{$t$--spanner} of $G$ if the lengths of shortest paths in $G$ are preserved in $H$ up to a multiplicative factor of $t$. The \emph{subsetwise spanner} problem aims to preserve distances in $G$ for only a subset of the vertices. We generalize the minimum-cost subsetwise spanner problem to one where vertices appear on multiple levels, which we call the \emph{multi-level graph spanner} (MLGS) problem, and describe two simple heuristics. Applications of this problem include road/network building and multi-level graph visualization, especially where vertices may require different grades of service. We formulate a 0--1 integer linear program (ILP) of size $O(|E||V{|}^2)$ for the more general minimum \emph{pairwise spanner problem}, which resolves an open question by Sigurd and Zachariasen on whether this problem admits a useful polynomial-size ILP. We extend this ILP formulation to the MLGS problem, and evaluate the heuristic and ILP performance on random graphs of up to 100 vertices and 500 edges.

This paper has been accepted in the Special Event on Analysis of Experimental Algorithms (SEA^2 2019)