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P-time Algorithms for Typical #EO Problems

Boning Meng, Juqiu Wang, Mingji Xia

TL;DR

The paper advances the complexity theory of counting within the Holant framework by establishing two complete dichotomies for $\#\textsf{EO}$: one for sets of binary and quaternary EO signatures (arity $\le 4$) and another for pure EO signatures. It introduces a polynomial-time algorithm for $0$- or $1$-rebalancing EO signature sets under a type condition, broadening tractable classes beyond the purely pure cases. The authors connect $\#\textsf{EO}$ to $\#\textsf{CSP}$ and the six-vertex model through gadget constructions and holographic transforms, pinpointing precise tractability boundaries and identifying challenging instances such as signatures $f_{40}$ and $f_{56}$. Together, these results push toward a complete classification of complex-weighted Holant problems and reveal structural mechanisms, like the passive receiving-sending process, that enable efficient counting.

Abstract

In this article, we study the computational complexity of counting weighted Eulerian orientations, denoted as \#\textsf{EO}. This problem is considered a pivotal scenario in the complexity classification for \textsf{Holant}, a counting framework of great significance. Our results consist of three parts. First, we prove a complexity dichotomy theorem for \#\textsf{EO} defined by a set of binary and quaternary signatures, which generalizes the previous dichotomy for the six-vertex model. Second, we prove a dichotomy for \#\textsf{EO} defined by a set of so-called pure signatures, which possess the closure property under gadget construction. Finally, we present a polynomial-time algorithm for \#\textsf{EO} defined by specific rebalancing signatures, which extends the algorithm for pure signatures to a broader range of problems, including \#\textsf{EO} defined by non-pure signatures such as $f_{40}$. We also construct a signature $f_{56}$ that is not rebalancing, and whether $\#\textsf{EO}(f_{56})$ is computable in polynomial time remains open.

P-time Algorithms for Typical #EO Problems

TL;DR

The paper advances the complexity theory of counting within the Holant framework by establishing two complete dichotomies for : one for sets of binary and quaternary EO signatures (arity ) and another for pure EO signatures. It introduces a polynomial-time algorithm for - or -rebalancing EO signature sets under a type condition, broadening tractable classes beyond the purely pure cases. The authors connect to and the six-vertex model through gadget constructions and holographic transforms, pinpointing precise tractability boundaries and identifying challenging instances such as signatures and . Together, these results push toward a complete classification of complex-weighted Holant problems and reveal structural mechanisms, like the passive receiving-sending process, that enable efficient counting.

Abstract

In this article, we study the computational complexity of counting weighted Eulerian orientations, denoted as \#\textsf{EO}. This problem is considered a pivotal scenario in the complexity classification for \textsf{Holant}, a counting framework of great significance. Our results consist of three parts. First, we prove a complexity dichotomy theorem for \#\textsf{EO} defined by a set of binary and quaternary signatures, which generalizes the previous dichotomy for the six-vertex model. Second, we prove a dichotomy for \#\textsf{EO} defined by a set of so-called pure signatures, which possess the closure property under gadget construction. Finally, we present a polynomial-time algorithm for \#\textsf{EO} defined by specific rebalancing signatures, which extends the algorithm for pure signatures to a broader range of problems, including \#\textsf{EO} defined by non-pure signatures such as . We also construct a signature that is not rebalancing, and whether is computable in polynomial time remains open.

Paper Structure

This paper contains 29 sections, 31 theorems, 22 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Counting weighted Eulerian orientation
  3. Our results
  4. Our methods
  5. Organization
  6. Preliminaries
  7. Definitions and notations
  8. Relationships between counting problems
  9. Gadget construction and signature matrix
  10. Two known dichotomies for #CSP and Six-vertex model
  11. Insights from tractable cases
  12. New notations and our results
  13. A set of binary or quaternary signatures
  14. Insights from tractable cases
  15. Proof outline of hardness result
  16. ...and 14 more sections

Key Result

Theorem 5

Suppose $\mathcal{F}$ is a finite set of $\textsf{EO}$ signatures with arity less or equal than 4. Then $\#\textsf{EO}(\mathcal{F})$ is either polynomial time computable or #P-hard. The classification criterion is explicit.

Figures (1)

  • Figure 1: An example of $G_X$. Blue edges refer to directed edges in $B$, heading from $S$ to $T$. Red edges refer to directed edges in $R$, heading from $T$ to $S$.

Theorems & Definitions (44)

  • Definition 1: #$\textsf{GH}$
  • Definition 2: $\#\textsf{CSP}$
  • Definition 3: $\textsf{Holant}$
  • Definition 4: $\#\textsf{EO}$
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Lemma 8
  • Lemma 9
  • Lemma 10: cai2020beyond
  • ...and 34 more