Exact Minimum Weight Spanners via Column Generation
Fritz Bökler, Markus Chimani, Henning Jasper, Mirko H. Wagner
TL;DR
This work tackles the exact minimum weight $α$-spanner problem, a task historically deemed impractical for large graphs, by comparing two exact ILP frameworks and delivering substantial engineering improvements to the path-based column generation approach. The authors show that a path-based model with tuned pricing (constrained shortest-path CSPs, multiobjective pricing via $\mu$-CSP and $\mu$-BiA$^*$, and robust initialization) scales to graphs with more than $16{,}000$ nodes in under $13$ minutes, often yielding tighter models than the arc-based multicommodity flow formulation. They demonstrate that, despite previous beliefs, exact approaches are viable in practice and enable direct evaluation of the strongest known heuristic on reasonably sized instances. The practical impact lies in enabling large-scale exact solutions for $α$-spanners and providing a rigorous basis for heuristic comparison in real-world graph settings.
Abstract
Given a weighted graph $G$, a minimum weight $α$-spanner is a least-weight subgraph $H\subseteq G$ that preserves minimum distances between all node pairs up to a factor of $α$. There are many results on heuristics and approximation algorithms, including a recent investigation of their practical performance [20]. Exact approaches, in contrast, have long been denounced as impractical: The first exact ILP (integer linear program) method [48] from 2004 is based on a model with exponentially many path variables, solved via column generation. A second approach [2], modeling via arc-based multicommodity flow, was presented in 2019. In both cases, only graphs with 40-100 nodes were reported to be solvable. In this paper, we briefly report on a theoretical comparison between these two models from a polyhedral point of view, and then concentrate on improvements and engineering aspects. We evaluate their performance in a large-scale empirical study. We report that our tuned column generation approach, based on multicriteria shortest path computations, is able to solve instances with over 16000 nodes within 13 minutes. Furthermore, now knowing optimal solutions for larger graphs, we are able to investigate the quality of the strongest known heuristic on reasonably sized instances for the first time.