非SNS符号模式的多项式时间奇异见证
Polynomial-Time Singular Witnesses for Non-SNS Sign Patterns
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中文总结 AI 辅助
本文针对非SNS符号模式，提出确定性多项式时间算法，可判定方阵符号模式是否为符号非奇异，或输出整数奇异见证，解决了《可满足性手册》中的相关猜想。
中文摘要 AI 辅助
符号非奇异性研究的是，给定任意具有指定元素符号的实矩阵是否非奇异。多项式时间算法可通过偶有向环的关联关系识别方阵的符号非奇异模式，但在模式为非符号非奇异的情况下，识别过程本身无法生成精确的数值见证。本文提出一种确定性多项式时间算法，对于任意方阵符号模式A，要么判定A为符号非奇异，要么输出B∈Z^(n×n)和z∈Z^n\{0}，满足sgn(B)=A且Bz=0。在归一化完美匹配后，一个偶有向环会产生两个符号相反的行列式项，使其中一项占优可得到行列式符号相反的端点实现，逐坐标改变其幅值会暴露出一个仿射变号步骤，该步骤的零点为有理数，清除其分母即可得到整数见证。B的元素有O(n²log n)位，z的元素有O(n³log n)位，该结果解决了《可满足性手册》中的猜想14.12.4。
英文摘要
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.
