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 $z\in\mathbb{Z}<sup>n\setminus{0}$ 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.

• The paper resolves a functional SNS-recognition problem by producing either a certificate that every matrix in a sign pattern is nonsingular or an explicit integer singular realization with a nonzero integer null vector.
• The method identifies opposite-sign determinant terms through even directed cycles, uses term domination to construct integer endpoint matrices, and applies coordinatewise affine interpolation to find a rational singularity with at most n²+1 determinant evaluations.
• The resulting witness has O(n² log n)-bit matrix entries and O(n³ log n)-bit null-vector entries, is efficiently verifiable, and yields consequences for polynomial-time autarky computation and NAESAT under stated deficiency conditions.

Overview

The paper resolves a functional version of the sign-nonsingularity (SNS) recognition problem for square sign patterns. Given $A\in\{-1,0,+1\}^{n\times n}$, the authors give a deterministic polynomial-time algorithm that either certifies that every real matrix in the qualitative class $Q(A)$ is nonsingular, or outputs an explicit integer witness of failure: matrices $B\in\mathbb{Z}^{n\times n}$ and $z\in\mathbb{Z}^n\setminus\{0\}$ with $\operatorname{sgn}(B)=A$ and $Bz=0$. Entries of $B$ have $O(n^2\log n)$ bits and entries of $z$ have $O(n^3\log n)$ bits. This settles Conjecture 14.12.4 of the Handbook of Satisfiability (2608.12075).

The contribution is deliberately framed as certificate extraction rather than new structure theory. Polynomial-time SNS recognition has been known since the structural work on even directed cycles by Seymour–Thomassen, Thomassen, Robertson–Seymour–Thomas, and McCuaig; what was missing is an efficient map from a non-SNS pattern to an exactly encoded singular member of its qualitative class.

Relation to previous work

The classical term criterion states that a square pattern is SNS if and only if it has at least one nonzero determinant term and all nonzero terms share one sign. Direct enumeration of terms is exponential, so the graph-theoretic line—convertible matrices, Pfaffian orientations, even directed cycles—replaced enumeration with structural recognition. The closest prior result is McCuaig's Theorem 4, which equates non-SNS with opposite-sign determinant terms, a real singular realization, and a row-balanced signed column scaling. That equivalence, however, is not functional: continuity on the segment between two opposite-sign endpoint realizations yields a real zero, but the determinant along that segment can have degree up to $Q(A)$0, so the zero need not be rational or admit a polynomial-bit exact encoding. Conversely, deriving a row-balanced scaling from a singular realization presupposes a kernel vector already in hand. The paper's new ingredient begins after an opposite-sign term is extracted: replacing the single high-degree segment by a coordinate path makes each step affine in one variable, forcing a rational zero recoverable from two integer determinants.

From even cycles to opposite-sign terms

After normalizing a perfect matching of the bipartite support graph to a positive diagonal (via column permutation and row sign flips), the authors build a signed digraph $Q(A)$1 from off-diagonal entries, subdividing each negative arc into a length-two path to obtain $Q(A)$2. A directed cycle of length $Q(A)$3 with $Q(A)$4 negative arcs contributes relative sign $Q(A)$5 to the identity term, which equals $Q(A)$6 exactly when the subdivided length $Q(A)$7 is even. Hence a determinant term opposite in sign to the identity exists if and only if $Q(A)$8 contains an even directed cycle, and such a cycle converts to an opposite-sign term in linear time. Even-cycle detection uses known algorithms as a black box; search reduces to decision by edge deletion, using at most $Q(A)$9 calls. If no perfect matching exists in the support graph, $B\in\mathbb{Z}^{n\times n}$0 itself is already an integer singular witness.

Constructing the integer witness

Two lemmas carry the construction. Term domination: setting $B\in\mathbb{Z}^{n\times n}$1, assigning magnitude $B\in\mathbb{Z}^{n\times n}$2 to the entries of one term and magnitude $B\in\mathbb{Z}^{n\times n}$3 elsewhere makes the selected term dominate all others, since $B\in\mathbb{Z}^{n\times n}$4, yielding integer realizations $B\in\mathbb{Z}^{n\times n}$5 with determinants of opposite signs. Coordinatewise interpolation: walking from $B\in\mathbb{Z}^{n\times n}$6 to $B\in\mathbb{Z}^{n\times n}$7 by changing one entry magnitude at a time, the determinant is affine on each edge; either some vertex determinant vanishes directly, or some edge exhibits a sign change with unique rational root $B\in\mathbb{Z}^{n\times n}$8. Clearing the single denominator gives an integer singular matrix in $B\in\mathbb{Z}^{n\times n}$9, using at most $z\in\mathbb{Z}^n\setminus\{0\}$0 exact determinant evaluations. A worked $z\in\mathbb{Z}^n\setminus\{0\}$1 example illustrates the full pipeline, producing $z\in\mathbb{Z}^n\setminus\{0\}$2 and the integer certificate $z\in\mathbb{Z}^n\setminus\{0\}$3 with null vector $z\in\mathbb{Z}^n\setminus\{0\}$4.

Bit complexity follows from Leibniz and Hadamard bounds: endpoint determinants have $z\in\mathbb{Z}^n\setminus\{0\}$5 bits, the reduced numerator and denominator of $z\in\mathbb{Z}^n\setminus\{0\}$6 inherit this bound, and exact elimination via Cramer's rule produces a null vector with $z\in\mathbb{Z}^n\setminus\{0\}$7-bit entries, i.e., $z\in\mathbb{Z}^n\setminus\{0\}$8 here. Notably, verification of the pair $z\in\mathbb{Z}^n\setminus\{0\}$9—checking signs, $\operatorname{sgn}(B)=A$0, and $\operatorname{sgn}(B)=A$1—is polynomial time and does not invoke the even-cycle machinery at all.

Consequences

Beyond resolving Handbook Conjecture 14.12.4, the paper records two corollaries obtained by combining the main theorem with Theorem 14.12.1 of the Handbook and Kullmann's matrix–SAT correspondence: for complement-invariant clause-sets with maximal reduced deficiency $\operatorname{sgn}(B)=A$2, a quasi-maximal autarky is computable and satisfiability decidable in polynomial time; equivalently, for clause-sets with $\operatorname{sgn}(B)=A$3, a quasi-maximal balanced autarky is computable and NAESAT decidable in polynomial time. In qualitative matrix analysis, the theorem supplies explicit polynomial-bit numerical data witnessing degeneracy whenever signs do not force nonsingularity.

Limitations and open questions

The paper is explicit about scope. It claims nothing for rectangular L-matrices or positive bounded deficiency—the corresponding extension is Conjecture 14.12.5 in the Handbook and remains open. The even-cycle machinery is used strictly as a black box, so any improvement in that front end transfers directly, but no new structural insight into even directed cycles is offered. The bit bounds are asymptotic; constant factors and practical performance of fraction-free elimination on the coordinate path are not analyzed.

Conclusion

The paper converts known combinatorial obstructions to sign-nonsingularity into exact algebraic certificates: term domination produces opposite-sign endpoints, coordinatewise affine interpolation forces a rational crossing, denominator clearing yields an integer singular realization, and exact elimination adds a short verifiable null vector. The principal open direction is extending certificate extraction from square patterns to rectangular patterns with fixed column excess.