有序图中的隐藏电路与精确计数

中文摘要 AI 辅助

我们证明了在多项式时间图灵归约下，在以下三类简单无权图上计算完美匹配是$\#P$-完全的：单调图、单位区间图和弦置换图。单调图的结果解决了Dyer、Jerrum和Müller（JACM 2017）留下的精确计数复杂性开放问题，并补充了他们的快速混合定理。受量子电路启发，我们的归约使用全局耦合的匹配传递算子实现电路模拟。关键构造是一个精确投影，通过多项式长度的归一化传递序列实现，该投影恢复张量积局部性并使编码门可组合。基于插值的消去随后将电路求值归约为这三类图中的无权完美匹配计数。我们还将Dyer和Müller的QChains类置于距离遗传图中，并给出后者一个$O(n^2)$次算术运算的计数算法，改进了Curticapean和Marx（SODA 2016）可获得的$O(n^4)$界。结合先前结果，这些进展完成了Dyer和Müller图（SIDMA 2019）中图类的精确计数分类。

英文摘要

We prove that counting perfect matchings is $\#P$-complete under polynomial-time Turing reductions on each of three classes of simple, unweighted graphs: monotone graphs, unit interval graphs, and chordal permutation graphs. The monotone result settles the exact-counting complexity left open by Dyer, Jerrum, and Müller (JACM 2017), complementing their rapid-mixing theorem. Inspired by quantum circuits, our reductions implement a circuit simulation using globally coupled matching-transfer operators. The key construction is an exact projection, implemented by a polynomial-length sequence of normalized transfers, that restores tensor-product locality and makes encoded gates composable. Interpolation-based cancellation then reduces circuit evaluation to unweighted perfect-matching counts in all three classes. We also place Dyer and Müller's class QChains within the distance-hereditary graphs and give an $O(n^2)$-arithmetic-operation counting algorithm for the latter, improving the $O(n^4)$ bound obtainable from Curticapean and Marx (SODA 2016). Together with prior results, these advances complete the exact-counting classification of the graph classes in Dyer and Müller's diagram (SIDMA 2019).

发表机构

• Institute of Software, Chinese Academy of Sciences（中国科学院软件研究所）
• University of Regensburg（雷根斯堡大学）

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