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Holant* Dichotomy on Domain Size 3: A Geometric Perspective

Jin-Yi Cai, Jin Soo Ihm

TL;DR

The paper resolves the complexity of Holant^*_3(F) for any set F of real-valued symmetric ternary signatures on domain {\mathsf{B}, \mathsf{G}, \mathsf{R}} by proving a complete dichotomy. Central to the result is a geometric viewpoint on tensor decompositions and their orbits under orthogonal holographic transformations, which organizes signatures into five tractable classes (A–E) and a hardness region. The authors show that, after a real orthogonal change of basis, the tractability of Holant^*_3(F) is determined by where F lies in these classes and by the structure of its interactions with binary signatures; if not in one of the tractable forms, the problem is #-hard. The work also develops a scalable framework, proving that tractability for a pair of ternary signatures, plus a geometric analysis of the signature space, extends to arbitrary sets of signatures, culminating in a unified set of dichotomy theorems that subsume prior Boolean-domain results and extend to domain size 3. This provides a foundational, geometry-driven understanding of higher-domain Holant problems with concrete algorithmic criteria for tractability.

Abstract

Holant problems are a general framework to study the computational complexity of counting problems. It is a more expressive framework than counting constraint satisfaction problems (CSP) which are in turn more expressive than counting graph homomorphisms (GH). In this paper, we prove the first complexity dichotomy of $\mathrm{Holant}_3(\mathcal{F})$ where $\mathcal{F}$ is an arbitrary set of symmetric, real valued constraint functions on domain size $3$. We give an explicit tractability criterion and prove that, if $\mathcal{F}$ satisfies this criterion then $\mathrm{Holant}_3(\mathcal{F})$ is polynomial time computable, and otherwise it is \#P-hard, with no intermediate cases. We show that the geometry of the tensor decomposition of the constraint functions plays a central role in the formulation as well as the structural internal logic of the dichotomy.

Holant* Dichotomy on Domain Size 3: A Geometric Perspective

TL;DR

The paper resolves the complexity of Holant^*_3(F) for any set F of real-valued symmetric ternary signatures on domain {\mathsf{B}, \mathsf{G}, \mathsf{R}} by proving a complete dichotomy. Central to the result is a geometric viewpoint on tensor decompositions and their orbits under orthogonal holographic transformations, which organizes signatures into five tractable classes (A–E) and a hardness region. The authors show that, after a real orthogonal change of basis, the tractability of Holant^*_3(F) is determined by where F lies in these classes and by the structure of its interactions with binary signatures; if not in one of the tractable forms, the problem is #-hard. The work also develops a scalable framework, proving that tractability for a pair of ternary signatures, plus a geometric analysis of the signature space, extends to arbitrary sets of signatures, culminating in a unified set of dichotomy theorems that subsume prior Boolean-domain results and extend to domain size 3. This provides a foundational, geometry-driven understanding of higher-domain Holant problems with concrete algorithmic criteria for tractability.

Abstract

Holant problems are a general framework to study the computational complexity of counting problems. It is a more expressive framework than counting constraint satisfaction problems (CSP) which are in turn more expressive than counting graph homomorphisms (GH). In this paper, we prove the first complexity dichotomy of where is an arbitrary set of symmetric, real valued constraint functions on domain size . We give an explicit tractability criterion and prove that, if satisfies this criterion then is polynomial time computable, and otherwise it is \#P-hard, with no intermediate cases. We show that the geometry of the tensor decomposition of the constraint functions plays a central role in the formulation as well as the structural internal logic of the dichotomy.

Paper Structure

This paper contains 44 sections, 48 theorems, 50 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions
  4. Holographic Transformation
  5. Notation
  6. Symmetric Tensor Rank
  7. Known Dichotomy Theorems
  8. Miscellaneous
  9. Statement of the Dichotomy Theorem
  10. Tractability
  11. Class A
  12. Class B
  13. Class C
  14. Class D
  15. Class E
  16. ...and 29 more sections

Key Result

Theorem 2.1

If there is a holographic transformation mapping signature grid $\Omega$ to $\Omega'$, then $\mathop{\mathrm{Holant}}\nolimits_{\Omega} = \mathop{\mathrm{Holant}}\nolimits_{\Omega'}$.

Figures (17)

  • Figure 1: Notation for expressing a symmetric ternary domain $3$ constraint functions. This notation can be extended for higher arity signatures by using a larger triangle.
  • Figure 2: Ternary constraint functions $\mathbf{F}_1$, $\mathbf{G}_1$, $\mathbf{H}_1$, and a binary constraint function $\mathbf{B}_1$.
  • Figure 3: Ternary constraint functions $\mathbf{F}_2$, $\mathbf{G}_2$ and bianry constraint functions $\mathbf{H}_2, \mathbf{B}_2$.
  • Figure 4: Gadget $\mathbf{G}^{\otimes 3} \mathbf{F}$
  • Figure 5: Visualization of type $\mathop{\mathrm{I}}\nolimits(a, b)$.
  • ...and 12 more figures

Theorems & Definitions (89)

  • Theorem 2.1: Valiant's Holant Theorem 10.1109/FOCS.2006.7
  • Corollary 2.2
  • Definition 2.3: Definition 4.1 of comon_symmetric_2008
  • Corollary 2.4: Corollary 4.4 of comon_symmetric_2008
  • Lemma 2.5: Lemma 5.1 of comon_symmetric_2008
  • Proposition 2.6
  • proof
  • Definition 2.7: Definition 2.9 in cai_complexity_2017
  • Theorem 2.8: Theorem 2.12 in cai_complexity_2017
  • Theorem 2.9: Theorem 3.1, 3.2 in cai_dichotomy_2013
  • ...and 79 more