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The $\text{FP}^\text{NP}$ versus #P dichotomy for #EO

Boning Meng, Juqiu Wang, Mingji Xia

TL;DR

An FPNP vs. #P dichotomy for #EO is presented, demonstrating that #EO defined by a signature set is either #P-hard or polynomial-time computable with a specific NP oracle.

Abstract

The complexity classification of the Holant problem has remained unresolved for the past fifteen years. Counting complex-weighted Eulerian orientation problems, denoted as #EO, is regarded as one of the most significant challenges to the comprehensive complexity classification of the Holant problem. This article presents an $\text{FP}^\text{NP}$ vs. #P dichotomy for #EO, demonstrating that #EO defined by a signature set is either #P-hard or polynomial-time computable with a specific NP oracle. This result provides a comprehensive complexity classification for #EO, and potentially leads to a dichotomy for the Holant problem. Furthermore, we derive three additional dichotomies related to the Holant problem from the dichotomy for #EO.

The $\text{FP}^\text{NP}$ versus #P dichotomy for #EO

TL;DR

An FPNP vs. #P dichotomy for #EO is presented, demonstrating that #EO defined by a signature set is either #P-hard or polynomial-time computable with a specific NP oracle.

Abstract

The complexity classification of the Holant problem has remained unresolved for the past fifteen years. Counting complex-weighted Eulerian orientation problems, denoted as #EO, is regarded as one of the most significant challenges to the comprehensive complexity classification of the Holant problem. This article presents an vs. #P dichotomy for #EO, demonstrating that #EO defined by a signature set is either #P-hard or polynomial-time computable with a specific NP oracle. This result provides a comprehensive complexity classification for #EO, and potentially leads to a dichotomy for the Holant problem. Furthermore, we derive three additional dichotomies related to the Holant problem from the dichotomy for #EO.

Paper Structure

This paper contains 29 sections, 50 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Introduction of the counting problems
  3. Significance of #EO
  4. Our results
  5. Our approaches
  6. Organization
  7. Preliminary
  8. Definitions and notations
  9. Gadget construction and signature matrix
  10. Previous results
  11. Detailed version of Theorem \ref{['thm:dicoceo']}
  12. Reduce $\#\textsf{EO}^c$ to $\#\textsf{EO}$
  13. A specific generating process
  14. Not a root of unity
  15. Infinitely many roots of unity
  16. ...and 14 more sections

Key Result

Theorem 2

Let $\mathcal{F}$ be a set of EO signatures. Then $\#\textsf{EO}(\mathcal{F})$ is either in $\text{FP}^\text{NP}$ or #P-hard.

Theorems & Definitions (68)

  • Definition 1: Holant, #EO and#CSP
  • Theorem 2
  • Definition 3: $\#\textsf{EO}$ problems
  • Definition 4: $\textsf{Holant}$ problems
  • Lemma 5
  • Lemma 6
  • Definition 7
  • Lemma 8
  • Lemma 9
  • Definition 10
  • ...and 58 more