A better tester for bipartiteness?

TL;DR

This paper improves bipartiteness testing algorithms by leveraging a conjecture about the structure of near-bipartite graphs, demonstrating that adaptive testing can outperform non-adaptive methods under certain conditions.

Contribution

The paper proves a generalized conjecture about induced subgraphs in regular graphs and uses it to develop a more efficient bipartiteness tester with fewer queries, assuming the conjecture holds.

Findings

A new proof for regular graphs supporting the conjecture.

An improved bipartiteness testing algorithm with query complexity less than O(1/ε^2).

Evidence that adaptivity enhances testing efficiency under the conjecture.

Abstract

Alon and Krivelevich (SIAM J. Discrete Math. 15(2): 211-227 (2002)) show that if a graph is {\epsilon}-far from bipartite, then the subgraph induced by a random subset of O(1/{\epsilon}) vertices is bipartite with high probability. We conjecture that the induced subgraph is {\Omega}~({\epsilon})-far from bipartite with high probability. Gonen and Ron (RANDOM 2007) proved this conjecture in the case when the degrees of all vertices are at most O({\epsilon}n). We give a more general proof that works for any d-regular (or almost d-regular) graph for arbitrary degree d. Assuming this conjecture, we prove that bipartiteness is testable with one-sided error in time O(1/{\epsilon}^c), where c is a constant strictly smaller than two, improving upon the tester of Alon and Krivelevich. As it is known that non-adaptive testers for bipartiteness require {\Omega}(1/{\epsilon}^2) queries (Bogdanov and Trevisan, CCC 2004), our result shows, assuming the conjecture, that adaptivity helps in testing bipartiteness.