The Complexity of Approximately Counting Retractions to Square-Free Graphs

Jacob Focke; Leslie Ann Goldberg; Stanislav Živný

doi:10.1145/3458040

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arXiv · 1907.02319

The Complexity of Approximately Counting Retractions to Square-Free Graphs

Jacob Focke·Leslie Ann Goldberg·Stanislav Živný

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Abstract

A retraction is a homomorphism from a graph $G$ to an induced subgraph $H$ of $G$ that is the identity on $H$. In a long line of research, retractions have been studied under various algorithmic settings. Recently, the problem of approximately counting retractions was considered. We give a complete trichotomy for the complexity of approximately counting retractions to all square-free graphs (graphs that do not contain a cycle of length $4$). It turns out there is a rich and interesting class of graphs for which this problem is complete in the class $\#\mathrm{BIS}$. As retractions generalise homomorphisms, our easiness results extend to the important problem of approximately counting homomorphisms. By giving new $\#\mathrm{BIS}$-easiness results we now settle the complexity of approximately counting homomorphisms for a whole class of non-trivial graphs which were previously unresolved.

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Jacob Focke, Leslie Ann Goldberg, Stanislav Živný. 2019-07-04. The Complexity of Approximately Counting Retractions to Square-Free Graphs. https://doi.org/10.1145/3458040

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