基于拟多态性视角的欧拉定向计数的线性规划算法
An LP Algorithm for Counting Eulerian Orientations Through the Lens of Quasi-polymorphism
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中文总结 AI 辅助
本文针对带权欧拉定向计数的FP^NP侧情况，提出基于线性规划松弛的多项式时间算法，解决其是否属于FP的问题，得到完整的FP与#P二分结果。
中文摘要 AI 辅助
带权欧拉定向计数问题（#EO）在Holant问题的复杂度分类程序中发挥关键作用。近期研究成果建立了#EO问题的FP^NP与#P困难性二分。该二分的易处理侧可由承认三元异或运算拟多态性的函数刻画，留下了FP^NP侧的这些情况是否实际属于FP的问题未解决。本文通过为FP^NP侧的所有情况提供多项式时间算法解决了该问题。因此，我们得到了带权欧拉定向计数的完整FP与#P二分，以及进一步的带奇数元签名的复值Holant问题的二分。我们的算法基于线性规划松弛，但以非标准方式使用它。我们没有证明该松弛是整性的并直接从最优LP解求解问题，而是将该松弛作为结构工具，将拟多态性条件提升为普通多态性条件。这揭示了约束函数的仿射局部结构，进而产生易处理性。
英文摘要
The weighted Eulerian orientation counting problem ($\#\mathrm{EO}$) plays a key role in the complexity classification program for Holant problems. A recent result established an $\mathrm{FP}^{\mathrm{NP}}$ versus $\#\mathrm{P}$-hard dichotomy for $\#\mathrm{EO}$ problems. The tractable side of this dichotomy can be characterized by functions admitting quasi-polymorphisms of the ternary XOR operation, leaving open whether these cases on the $\mathrm{FP}^{\mathrm{NP}}$ side are in fact in FP. In this paper, we settle this question by giving a polynomial-time algorithm for all cases on the $\mathrm{FP}^{\mathrm{NP}}$ side. Consequently, we obtain a complete FP versus $\#\mathrm{P}$ dichotomy for counting weighted Eulerian orientations, and further for complex-valued Holant problems with an odd-arity signature. Our algorithm is based on a linear programming relaxation, but we use it in a nonstandard way. Instead of proving that the relaxation is integral and solving the problem directly from an optimal LP solution, we use the relaxation as a structural tool to lift the quasi-polymorphism condition to an ordinary polymorphism condition. This reveals an affine local structure of the constraint functions, which leads to tractability.
