The Complexity of Approximately Counting Retractions

TL;DR

This paper investigates the computational complexity of approximately counting graph retractions, providing a complete classification for certain graph classes and contrasting it with the known results for exact counting, revealing new complexity separations.

Contribution

It offers a complete trichotomy for approximately counting retractions to graphs with girth at least 5 and situates this problem within the broader landscape of approximate counting complexities.

Findings

Complete trichotomy for graphs of girth ≥ 5

Approximate counting of retractions is distinct from homomorphisms and list homomorphisms

Approximate counting of retractions is at least as hard as counting surjective homomorphisms and compactions

Abstract

Let $G$ be a graph that contains an induced subgraph $H$. A retraction from $G$ to $H$ is a homomorphism from $G$ to $H$ that is the identity function on $H$. Retractions are very well-studied: Given $H$, the complexity of deciding whether there is a retraction from an input graph $G$ to $H$ is completely classified, in the sense that it is known for which $H$ this problem is tractable (assuming $\mathrm{P}\neq \mathrm{NP}$). Similarly, the complexity of (exactly) counting retractions from $G$ to $H$ is classified (assuming $\mathrm{FP}\neq \#\mathrm{P}$). However, almost nothing is known about approximately counting retractions. Our first contribution is to give a complete trichotomy for approximately counting retractions to graphs of girth at least $5$. Our second contribution is to locate the retraction counting problem for each $H$ in the complexity landscape of related approximate counting problems. Interestingly, our results are in contrast to the situation in the exact counting context. We show that the problem of approximately counting retractions is separated both from the problem of approximately counting homomorphisms and from the problem of approximately counting list homomorphisms --- whereas for exact counting all three of these problems are interreducible. We also show that the number of retractions is at least as hard to approximate as both the number of surjective homomorphisms and the number of compactions. In contrast, exactly counting compactions is the hardest of all of these exact counting problems.