Year: 2026
Type: preprint
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}).
Source: arXiv (Cornell University)
Authors: Fei, Yumou, Rubinfeld, Ronitt
Institution: Massachusetts Institute of Technology
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W7164544318
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