Bipartite Density: From Mixing Time to Local Algorithms for Dense Subgraphs

Abstract

Classical spectral graph theory shows that edge conductance characterizes the optimal O(log n) mixing time of d-regular graphs when d is constant. When d grows with n, however, the optimal mixing time is O(log_d n), and this characterization no longer applies. We give a new condition based on bipartite density that provides a more refined combinatorial characterization of mixing time across different degree regimes. Using this new connection between density and mixing time, we revisit the local algorithmic approach to finding dense subgraphs and show that it can be sharpened and extended considerably.

(1) We improve Andersen's bicriteria approximation algorithms for finding dense bipartite subgraphs, both in approximation ratio and in output size. Our result can be interpreted as a local version of Bilu and Linial’s converse of the expander mixing lemma. The approximation guarantee improves in the high density regime, resembling Cheeger’s inequality in the high conductance regime.

(2) We provide the first tradeoff between the approximation guarantee and the output size for finding small dense bipartite subgraphs. This tradeoff interpolates between the improved bicriteria guarantee above and a true approximation algorithm with no loss in the output size. In the high density regime, the true approximation guarantee has a subpolynomial approximation ratio.

(3) We extend this approach to the densest k-subgraph problem, answering a question raised by Andersen. This gives the first local approximation algorithm for the densest k-subgraph problem. Moreover, our approach identifies a new high density regime, distinct from the previous almost clique regime, where a polynomial time algorithm achieves a subpolynomial approximation ratio.

This analysis also gives a new random walk proof of the converse of the expander mixing lemma. Our algorithmic results can be viewed as a realization of Bilu and Linial’s speculation that their spectral bound might be useful in designing graph partitioning algorithms.