An LP Algorithm for Counting Eulerian Orientations Through the Lens of Quasi-polymorphism

2026-07-30

AI Summary

This work resolves the long-standing open question of whether the counting problem of weighted Eulerian orientations (#EO) lies in FP when it is known to be in FP^NP, thereby completing its dichotomy between FP and #P. The key innovation lies in the novel use of linear programming relaxations not as a direct solving mechanism but as a structural analysis tool, which enables the elevation of quasi-polymorphisms to genuine polymorphisms and reveals an affine local structure inherent in the constraint functions. Leveraging this insight, the authors devise a polynomial-time algorithm for #EO and further establish a complete FP/#P dichotomy for complex-valued Holant problems involving signatures of odd arity.

Abstract

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.

Problem

• Eulerian orientations
• counting complexity
• Holant problems
• dichotomy
• quasi-polymorphism

Innovation

• linear programming relaxation
• quasi-polymorphism
• Eulerian orientations
• Holant problems
• affine structure