eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-01-12
41:1
41:16
10.4230/LIPIcs.ITCS.2018.41
article
Graph Clustering using Effective Resistance
Alev, Vedat Levi
Anari, Nima
Lau, Lap Chi
Oveis Gharan, Shayan
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.
sum(i < j) |E(V(i), V(j)| < w(E)/\delta
and the effective resistance diameter of each of the induced subgraphs
G[V(i)] is at most \delta^3 times the inverse of the average weighted degree, i.e.
max{ Reff(u, v) : u, v \in V(i)} < \delta^3 · |V|/w(E)
for all i = 1,..., h. In particular, it is possible to remove one
percent of weight of edges of any given graph such that each of the
resulting connected components has effective resistance diameter at
most the inverse of the average weighted degree. Our proof is based
on a new connection between effective resistance and low conductance
sets. 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol094-itcs2018/LIPIcs.ITCS.2018.41/LIPIcs.ITCS.2018.41.pdf
Electrical Flows
Effective Resistance
Conductance
Graph Partitioning