A deterministic algorithm for signing bipartite graphs at the Ramanujan bound

2026-10-07

AI Summary

This study addresses the Bilu-Linial signing problem for bipartite graphs, aiming to construct sign assignments such that the operator norm of the signed adjacency matrix is strictly bounded below the Ramanujan threshold. Methodologically, this work proposes a polynomial-time deterministic algorithm that achieves optimal spectral control through regularized sign selection and a recursive vertex deletion mechanism. Its core innovation lies in transforming a randomized recursive repair framework into a fully deterministic procedure, thereby overcoming the limitations of existing randomized algorithms. Ultimately, this approach successfully constructs signing matrices for maximum-degree-bounded bipartite graphs that satisfy strict Ramanujan bound constraints, achieving a precise approximation of the theoretical lower bound.

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 $Δ\ge3$, the algorithm assigns signs to its edges so that the signed adjacency matrix has operator norm strictly less than $2\sqrt{Δ-1}$. 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.

Problem

• bipartite graphs
• Bilu-Linial signing problem
• Ramanujan bound
• signed adjacency matrix
• operator norm

Innovation

• Deterministic algorithm
• Bipartite graphs
• Bilu-Linial signing
• Ramanujan bound
• Signed adjacency matrix