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From an odd arity signature to a Holant dichotomy

Boning Meng, Juqiu Wang, Mingji Xia, Jiayi Zheng

TL;DR

This work advances the classification of complex-valued Holant problems on the Boolean domain by proving a dichotomy for Holant^odd, contingent on the existence of a non-trivial odd-arity signature. Central to the approach is a generalized decomposition lemma that either reduces a Holant instance to a tensor product f⊗g or places the problem in FP^NP via #EO dichotomy-guided reductions. The authors develop a robust toolkit—gadget construction with signature matrices, holographic transformations including SLOCC, polynomial interpolation, and UPF-based factorization—to realize and reduce signatures while controlling complexity. The results unify and extend prior dichotomies, provide a principled path for reductions in complex-valued Holant, and point to the remaining challenge of the even-arity irreducible case as a direction for future work with potential implications for broader counting complexity landscapes.

Abstract

\textsf{Holant} is an essential framework in the field of counting complexity. For over fifteen years, researchers have been clarifying the complexity classification for complex-valued \textsf{Holant} on the Boolean domain, a challenge that remains unresolved. In this article, we prove a complexity dichotomy for complex-valued \textsf{Holant} on Boolean domain when a non-trivial signature of odd arity exists. This dichotomy is based on the dichotomy for \textsf{\#EO}, and consequently is an $\text{FP}^\text{NP}$ vs. \#P dichotomy as well, stating that each problem is either in $\text{FP}^\text{NP}$ or \#P-hard. Furthermore, we establish a generalized version of the decomposition lemma for complex-valued \textsf{Holant} on Boolean domain. It asserts that each signature can be derived from its tensor product with other signatures, or conversely, the problem itself is in $\text{FP}^\text{NP}$. We believe that this result is a powerful method for building reductions in complex-valued \textsf{Holant}, as it is also employed as a pivotal technique in the proof of the aforementioned dichotomy in this article.

From an odd arity signature to a Holant dichotomy

TL;DR

This work advances the classification of complex-valued Holant problems on the Boolean domain by proving a dichotomy for Holant^odd, contingent on the existence of a non-trivial odd-arity signature. Central to the approach is a generalized decomposition lemma that either reduces a Holant instance to a tensor product f⊗g or places the problem in FP^NP via #EO dichotomy-guided reductions. The authors develop a robust toolkit—gadget construction with signature matrices, holographic transformations including SLOCC, polynomial interpolation, and UPF-based factorization—to realize and reduce signatures while controlling complexity. The results unify and extend prior dichotomies, provide a principled path for reductions in complex-valued Holant, and point to the remaining challenge of the even-arity irreducible case as a direction for future work with potential implications for broader counting complexity landscapes.

Abstract

\textsf{Holant} is an essential framework in the field of counting complexity. For over fifteen years, researchers have been clarifying the complexity classification for complex-valued \textsf{Holant} on the Boolean domain, a challenge that remains unresolved. In this article, we prove a complexity dichotomy for complex-valued \textsf{Holant} on Boolean domain when a non-trivial signature of odd arity exists. This dichotomy is based on the dichotomy for \textsf{\#EO}, and consequently is an vs. \#P dichotomy as well, stating that each problem is either in or \#P-hard. Furthermore, we establish a generalized version of the decomposition lemma for complex-valued \textsf{Holant} on Boolean domain. It asserts that each signature can be derived from its tensor product with other signatures, or conversely, the problem itself is in . We believe that this result is a powerful method for building reductions in complex-valued \textsf{Holant}, as it is also employed as a pivotal technique in the proof of the aforementioned dichotomy in this article.

Paper Structure

This paper contains 20 sections, 55 theorems, 19 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions and notations
  4. Frameworks of counting problems
  5. Fundamental methods
  6. Gadget construction and signature matrix
  7. Holographic transformations and SLOCC
  8. Polynomial interpolation
  9. Signature factorization
  10. Known dichotomies
  11. #CSP and $\textup{\#CSP}_d$
  12. $\textup{Holant}^c$
  13. #EO
  14. Main results
  15. Decomposition lemma
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\mathcal{F}$ be a set of signatures and $f$, $g$ be two signatures. Then one of the following holds.

Figures (2)

  • Figure 1: The reduction map for Holant with $\Delta_0$. We use the term 'A$\to$B' to mean the corresponding reduction in our proof. It is noteworthy that a node may have multiple outgoing arcs, and in this case at least one of the reduction holds.
  • Figure 2: The reduction map for $\textup{Holant}^\text{odd}$. We use the term 'A$\to$B' to mean the corresponding reduction in our proof. It is noteworthy that a node may have multiple outgoing arcs, and in this case at least one of the reduction holds.

Theorems & Definitions (68)

  • Theorem 1: Decomposition lemma
  • Theorem 2: Dichotomy for $\textup{Holant}^\text{odd}$
  • Definition 3: $\textup{Holant}$ problems
  • Definition 4: $\#\textup{EO}$ problems cai2020beyond
  • Lemma 5: cai2020beyond
  • Definition 6: $\#\textup{CSP}$
  • Definition 7
  • Lemma 8: cai2017complexity
  • Lemma 9
  • Lemma 10: cai2020holantoddarity
  • ...and 58 more