Vanishing Signatures, Orbit Closure, and the Converse of the Holant Theorem

2025-09-13

AI Summary

This paper investigates the converse of the Holant theorem: if two sets of tensors are Holant-indistinguishable, must they be related by a holographic transformation? It focuses on whether “vanishing signatures” constitute the sole obstruction. Leveraging algebraic invariant theory, the authors analyze orbit closure intersections under the action of the general linear group GL_q, integrating insights from holographic approximation and tensor network contraction. They formulate and prove two approximate converse theorems, establishing that vanishing signatures are indeed the only barrier to the exact converse. Furthermore, they provide the first complete characterization of indistinguishability for bounded-degree graph homomorphisms: on graphs of maximum degree ≤ 3, two graphs with invertible adjacency matrices are graph-isomorphic if and only if they are Holant-indistinguishable. Finally, they show that this Holant indistinguishability problem is computationally equivalent in hardness to the graph isomorphism problem.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Valiant's Holant theorem is a powerful tool for algorithms and reductions for counting problems. It states that if two sets $mathcal{F}$ and $mathcal{G}$ of tensors (a.k.a. constraint functions or signatures) are related by a emph{holographic transformation}, then $mathcal{F}$ and $mathcal{G}$ are emph{Holant-indistinguishable}, i.e., every tensor network using tensors from $mathcal{F}$, resp. from $mathcal{G}$, contracts to the same value. Xia (ICALP 2010) conjectured the converse of the Holant theorem, but a counterexample was found based on emph{vanishing} signatures, those which are Holant-indistinguishable from 0. We prove two near-converses of the Holant theorem using techniques from invariant theory. (I) Holant-indistinguishable $mathcal{F}$ and $mathcal{G}$ always admit two sequences of holographic transformations mapping them arbitrarily close to each other, i.e., their $ ext{GL}_q$-orbit closures intersect. (II) We show that vanishing signatures are the only true obstacle to a converse of the Holant theorem. As corollaries of the two theorems we obtain the first characterization of homomorphism-indistinguishability over graphs of bounded degree, a long standing open problem, and show that two graphs with invertible adjacency matrices are isomorphic if and only if they are homomorphism-indistinguishable over graphs with maximum degree at most three. We also show that Holant-indistinguishability is complete for a complexity class extbf{TOCI} introduced by Lysikov and Walter, and hence hard for graph isomorphism.

Problem

• Proving near-converses of Valiant's Holant theorem
• Characterizing homomorphism-indistinguishability over bounded-degree graphs
• Identifying vanishing signatures as obstacles to Holant converse

Innovation

• Using invariant theory techniques
• Proving near-converses of Holant theorem
• Characterizing homomorphism-indistinguishability over bounded-degree graphs