Approximation Algorithms and an Integer Program for Multi-level Graph Spanners

• Published in: Analysis of Experimental Algorithms Vol. 11544; pp. 541 - 562
• Main Authors: Ahmed, Reyan, Hamm, Keaton, Latifi Jebelli, Mohammad Javad, Kobourov, Stephen, Sahneh, Faryad Darabi, Spence, Richard
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2019; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Graph spanners; Integer programming; Multi-level graph representation
• ISBN: 9783030340285; 3030340287
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-030-34029-2_35

Given a weighted graph G(V, E) and $$t \ge 1$$ , a subgraph H is a t–spanner of G if the lengths of shortest paths in G are preserved in H up to a multiplicative factor of t. The 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 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 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.