Towards Resistance Sparsifiers

Dinitz, Michael; Krauthgamer, Robert; Wagner, Tal

doi:10.4230/LIPIcs.APPROX-RANDOM.2015.738

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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.738
URN: urn:nbn:de:0030-drops-53334

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Michael Dinitz

Robert Krauthgamer

Tal Wagner

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Michael Dinitz, Robert Krauthgamer, and Tal Wagner. Towards Resistance Sparsifiers. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 738-755, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.738

Abstract

We study resistance sparsification of graphs, in which the goal is to find a sparse subgraph (with reweighted edges) that approximately preserves the effective resistances between every pair of nodes. We show that every dense regular expander admits a (1+epsilon)-resistance sparsifier of size ~O(n/epsilon), and conjecture this bound holds for all graphs on n nodes. In comparison, spectral sparsification is a strictly stronger notion and requires Omega(n/epsilon^2) edges even on the complete graph. Our approach leads to the following structural question on graphs: Does every dense regular expander contain a sparse regular expander as a subgraph? Our main technical contribution, which may of independent interest, is a positive answer to this question in a certain setting of parameters. Combining this with a recent result of von Luxburg, Radl, and Hein (JMLR, 2014) leads to the aforementioned resistance sparsifiers.

Keywords

edge sparsification
spectral sparsifier
graph expansion
effective resistance
commute time

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