Polynomial-Time Singular Witnesses for Non-SNS Sign Patterns

Abstract

Sign-nonsingularity asks whether every real matrix with prescribed entry signs is nonsingular. Polynomial-time algorithms recognize square sign-nonsingular patterns through their connection with even directed cycles, but recognition does not itself produce an exact numerical witness in the negative case. We give a deterministic polynomial-time algorithm that, for any square sign pattern $A$, either reports that $A$ is sign-nonsingular or outputs $B\in\mathbb{Z}^{n\times n}$ and $z\in\mathbb{Z}^n\setminus\{0\}$ such that $\operatorname{sgn}(B)=A$ and $Bz=0$. After normalizing a perfect matching, an even directed cycle yields two determinant terms of opposite signs. Making either term dominant produces endpoint realizations with opposite determinant signs. Changing their magnitudes one coordinate at a time exposes an affine sign-changing step, whose zero is rational; clearing its denominator gives the integer witness. Entries of $B$ have $O(n^2\log n)$ bits, and entries of $z$ have $O(n^3\log n)$ bits. The result settles Conjecture 14.12.4 in the Handbook of Satisfiability.