{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article10584","name":"Graph Clustering using Effective Resistance","abstract":"We design a polynomial time algorithm that for any weighted undirected graph G = (V, E, w) and sufficiently large \\delta > 1, partitions V into subsets V(1),..., V(h) for some h>= 1, such that at most \\delta^{-1} fraction of the weights are between clusters, i.e.\r\n\r\nsum(i < j) |E(V(i), V(j)| < w(E)\/\\delta \r\n\r\nand the effective resistance diameter of each of the induced subgraphs\r\nG[V(i)] is at most \\delta^3 times the inverse of the average weighted degree, i.e.\r\n\r\nmax{ Reff(u, v) : u, v \\in V(i)} < \\delta^3 \u00b7 |V|\/w(E)\r\n\r\nfor all i = 1,..., h. In particular, it is possible to remove one\r\npercent of weight of edges of any given graph such that each of the\r\nresulting connected components has effective resistance diameter at\r\nmost the inverse of the average weighted degree. Our proof is based\r\non a new connection between effective resistance and low conductance\r\nsets. We show that if the effective resistance between two vertices u and v is large, then there must be a low conductance cut separating u from v. This implies that very mildly expanding graphs have constant effective resistance diameter. We believe that this connection could be of independent interest in algorithm design.","keywords":["Electrical Flows","Effective Resistance","Conductance","Graph Partitioning"],"author":[{"@type":"Person","name":"Alev, Vedat Levi","givenName":"Vedat Levi","familyName":"Alev"},{"@type":"Person","name":"Anari, Nima","givenName":"Nima","familyName":"Anari"},{"@type":"Person","name":"Lau, Lap Chi","givenName":"Lap Chi","familyName":"Lau"},{"@type":"Person","name":"Oveis Gharan, Shayan","givenName":"Shayan","familyName":"Oveis Gharan"}],"position":41,"pageStart":"41:1","pageEnd":"41:16","dateCreated":"2018-01-12","datePublished":"2018-01-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Alev, Vedat Levi","givenName":"Vedat Levi","familyName":"Alev"},{"@type":"Person","name":"Anari, Nima","givenName":"Nima","familyName":"Anari"},{"@type":"Person","name":"Lau, Lap Chi","givenName":"Lap Chi","familyName":"Lau"},{"@type":"Person","name":"Oveis Gharan, Shayan","givenName":"Shayan","familyName":"Oveis Gharan"}],"copyrightYear":"2018","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.ITCS.2018.41","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/arxiv.org\/abs\/1711.06530","http:\/\/dx.doi.org\/10.1145\/990308.990313"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6297","volumeNumber":94,"name":"9th Innovations in Theoretical Computer Science Conference (ITCS 2018)","dateCreated":"2018-01-12","datePublished":"2018-01-12","editor":{"@type":"Person","name":"Karlin, Anna R.","givenName":"Anna R.","familyName":"Karlin"},"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article10584","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6297"}}}