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

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-c...

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Bibliographic Details
Published inAnalysis 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
LanguageEnglish
Published Switzerland Springer International Publishing AG 2019
Springer International Publishing
SeriesLecture Notes in Computer Science
Subjects
Graph spanners
Integer programming
Multi-level graph representation
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ISBN9783030340285
3030340287
ISSN0302-9743
1611-3349
DOI10.1007/978-3-030-34029-2_35

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Summary: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.
Bibliography:This work was supported in part by NSF grants CCF-1740858, CCF-1712119, and DMS-1839274.
Original Abstract: Given a weighted graph G(V, E) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \ge 1$$\end{document}, 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(|E||V|^2)$$\end{document} 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.
ISBN:9783030340285
3030340287
ISSN:0302-9743
1611-3349
DOI:10.1007/978-3-030-34029-2_35