A deterministic polynomial-time algorithm for the Bilu--Linial signing problem on bipartite graphs

Abstract

We give a deterministic polynomial-time algorithm for the Bilu--Linial signing problem on bipartite graphs. For every finite simple bipartite graph of maximum degree at most an integer $\Delta \ge 3$, the algorithm assigns signs to its edges so that the signed adjacency matrix has operator norm strictly less than $2\sqrt{\Delta -1}$. Our algorithm builds on the randomized recursive repair framework of Jadbabaie, Saberi, and Sra~\cite{JSS26}, with deterministic rules for sign selection and vertex deletion.

arXiv comment20 pages. We have improved the presentation and clarified some details of the arguments