The Converse of the Real Orthogonal Holant Theorem

TL;DR

This paper proves a highly general version of the converse of the Real Orthogonal Holant Theorem, establishing that Holant-indistinguishable real signatures are orthogonally equivalent, resolving a conjecture and impacting various areas in complexity and tensor analysis.

Contribution

It establishes that Holant-indistinguishable real signatures are orthogonally equivalent, confirming a conjecture and extending the theorem's implications to graph isomorphism, matrix similarity, and tensor decomposition.

Findings

Holant-indistinguishable signatures are orthogonally equivalent.

Resolved a conjecture of Xia (2010) on the converse of the Holant theorem.

Characterized orthogonally decomposable sets of symmetric tensors.

Abstract

The Holant theorem is a powerful tool for studying the computational complexity of counting problems in the Holant framework. Due to the great expressiveness of the Holant framework, a converse to the Holant theorem would itself be a very powerful counting indistinguishability theorem. The most general converse does not hold, but we prove the following, still highly general, version: if any two sets of real-valued signatures are Holant-indistinguishable, then they are equivalent up to an orthogonal transformation. This resolves a partially open conjecture of Xia (2010). Consequences of this theorem include the well-known result that homomorphism counts from all graphs determine a graph up to isomorphism, the classical sufficient condition for simultaneous orthogonal similarity of sets of real matrices, and a combinatorial characterization of simultaneosly orthogonally decomposable (odeco) sets of symmetric tensors.