An Algorithmic Meta Theorem for Homomorphism Indistinguishability

TL;DR

This work provides a broad algorithmic framework for deciding homomorphism indistinguishability: for any k-recognisable graph class 𝔽 of treewidth at most k−1, Hom-Ind(𝔽) is decidable in randomized polynomial time; this extends to CMSO2-specified classes and to pathwidth-bounded cases with deterministic polynomial-time or XP algorithms. The authors develop a graph-algebraic approach using labelled graphs and homomorphism tensors, leveraging Courcelle’s recognisability and finite-dimensional state spaces to certify indistinguishability. They further connect the framework to the Lasserre hierarchy and show that level-t Lasserre feasibility corresponds to homomorphism indistinguishability over a minor-closed class 𝔏_t, with a randomised polynomial-time algorithm for fixed t. Lower bounds are established via WL indistinguishability and HomIndSize, yielding coNP- and coW[1]-hardness results that illustrate intrinsic complexity limits and the role of treewidth. The results collectively provide a principled, meta-theorem tying recognisability, graph algebras, and logical definability to tractable decision procedures for a broad family of graph equivalences."

Abstract

Two graphs $G$ and $H$ are homomorphism indistinguishable over a family of graphs $\mathcal{F}$ if for all graphs $F \in \mathcal{F}$ the number of homomorphisms from $F$ to $G$ is equal to the number of homomorphism from $F$ to $H$. Many natural equivalence relations comparing graphs such as (quantum) isomorphism, cospectrality, and logical equivalences can be characterised as homomorphism indistinguishability relations over various graph classes. For a fixed graph class $\mathcal{F}$, the decision problem HomInd($\mathcal{F}$) asks to determine whether two input graphs $G$ and $H$ are homomorphism indistinguishable over $\mathcal{F}$. The problem HomInd($\mathcal{F}$) is known to be decidable only for few graph classes $\mathcal{F}$. We show that HomInd($\mathcal{F}$) admits a randomised polynomial-time algorithm for every graph class $\mathcal{F}$ of bounded treewidth which is definable in counting monadic second-order logic CMSO2. Thereby, we give the first general algorithm for deciding homomorphism indistinguishability. This result extends to a version of HomInd where the graph class $\mathcal{F}$ is specified by a CMSO2-sentence and a bound $k$ on the treewidth, which are given as input. For fixed $k$, this problem is randomised fixed-parameter tractable. If $k$ is part of the input then it is coNP- and coW[1]-hard. Addressing a problem posed by Berkholz (2012), we show coNP-hardness by establishing that deciding indistinguishability under the $k$-dimensional Weisfeiler--Leman algorithm is coNP-hard when $k$ is part of the input.

Figures (2)

• Figure 1: The (bi)labelled generators of $\mathcal{TW}(k)$ in wire notion of mancinska_quantum_2020. A vertex carries in-label (out-label) $i$ if it is connected to the index $i$ on the left (right) by a wire. Actual edges and vertices of the graph are depicted in black.
• Figure 2: Representatives for $\sim_{\mathcal{W}}$ for $k=1$ and $\mathcal{W}$ the class of paths from \ref{['ex:paths']}.

Theorems & Definitions (65)

• Theorem 1.1: restate=mainRP, label=
• Theorem 1.2: restate=mainFPT, label=
• Theorem 1.3: restate=mainPW, label=
• Theorem 1.4: restate=mainD, label=
• Theorem 1.5: restate=mainC, label=
• proof
• Lemma 2.2: bodlaender_partial_1998
• Lemma 2.3
• proof
• Definition 2.4