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.
@InProceedings{dinitz_et_al:LIPIcs.APPROX-RANDOM.2015.738, author = {Dinitz, Michael and Krauthgamer, Robert and Wagner, Tal}, title = {{Towards Resistance Sparsifiers}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {738--755}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.738}, URN = {urn:nbn:de:0030-drops-53334}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.738}, annote = {Keywords: edge sparsification, spectral sparsifier, graph expansion, effective resistance, commute time} }
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