An Algorithmic Meta Theorem for Homomorphism Indistinguishability
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914678843637760 |
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| author | Seppelt, Tim |
| author_facet | Seppelt, Tim |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08989 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An Algorithmic Meta Theorem for Homomorphism Indistinguishability Seppelt, Tim Logic in Computer Science Computational Complexity Discrete Mathematics Combinatorics 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. |
| title | An Algorithmic Meta Theorem for Homomorphism Indistinguishability |
| topic | Logic in Computer Science Computational Complexity Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2402.08989 |