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arXiv · 2503.21194

Matchgate signatures under variable permutations

Boning Meng·Yicheng Pan
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Abstract

In this article, we give a sufficient and necessary condition for determining whether a matchgate signature retains its property under a certain variable permutation, which can be checked in polynomial time. We also define the concept of permutable matchgate signatures, and use it to erase the gap between Pl-\#CSP and \#CSP on planar graphs in the previous study. We provide a detailed characterization of permutable matchgate signatures as well, by presenting their relation to symmetric matchgate signatures. In addition, we prove a dichotomy for Pl-$\#R_D$-CSP where $D\ge 3$ is an integer.

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Boning Meng, Yicheng Pan. 2025-03-27. Matchgate signatures under variable permutations. https://arxiv.org/abs/2503.21194

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