Testing Bipartiteness in Logarithmic Rounds

2026-06-11

AI Summary

This work addresses the problem of testing bipartiteness in bounded-degree graphs with the goal of reducing both the number and length of random walks required. By introducing, for the first time in this context, the Goemans–Williamson semidefinite programming relaxation for Max-Cut, the authors improve upon the classic Goldreich–Ron algorithm. Their approach achieves bipartiteness testing using only $O(\sqrt{n})$ random walks, each of length $O(\log n)$, significantly lowering the query complexity. Furthermore, this technique yields a round-optimal streaming algorithm that decides bipartiteness in $O(\log n)$ passes using $O(\sqrt{n} \log n)$ space. The number of passes matches the theoretical lower bound and improves upon the previous best result, which required $O(\sqrt{n \log n})$ random walks.

Abstract

The seminal work of Goldreich and Ron (\textit{Combinatorica, 1999}) showed that bipartiteness of bounded-degree graphs can be tested using $O(\sqrt{n\log n})$ random walks of length $O(\log^{6} n)$. In this work, we improve their result by showing that $O(\sqrt{n})$ random walks of length $O(\log n)$ suffice. As a corollary, we obtain an $O(\log n)$-pass, $O(\sqrt{n}\log n)$-space streaming algorithm for testing bipartiteness, whose pass complexity is optimal in light of a recent lower bound of Fei, Minzer, and Wang (\textit{arXiv, 2026}). Our proof takes a different approach from that of Goldreich and Ron, using the semidefinite programming relaxation for Max-Cut introduced by Goemans and Williamson (\textit{J. ACM, 1995}).

Problem

• bipartiteness
• graph testing
• random walks
• streaming algorithm
• logarithmic rounds

Innovation

• bipartiteness testing
• random walks
• streaming algorithm
• semidefinite programming
• Max-Cut relaxation