The Complexity of Approximately Counting Retractions

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

The complexity of approximately counting retractions

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 P ≠ NP). Similarly, the complexity of (exactly) counting retractions from G to H is classified (assuming FP ≠ #P). However, almost nothing is known about approximately counting retractions. Our first contribution is to give a complete trichotomy for approximately counting retractions to trees. The result is as follows: (1) Approximately counting retractions to a tree H is in FP if H is a star, a single looped vertex, or an edge with two loops. (2) Otherwise, if H is an irreflexive caterpillar or a partially bristled reflexive path, then approximately counting retractions to H is equivalent to approximately counting the independent sets of a bipartite graph — a problem which is complete in the approximate counting complexity class RHπ1. (3) Finally, if none of these hold, then approximately counting retractions to H is #P-complete under approximation-preserving reductions. Our second contribution is to locate the retraction counting problem 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 these problems

The complexity of approximately counting retractions

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 P ≠ NP). Similarly, the complexity of (exactly) counting retractions from G to H is classified (assuming FP ≠ #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 without short cycles. The result is as follows: (1) Approximately counting retractions to a graph H of girth at least 5 is in FP if every connected component of H is a star, a single looped vertex, or an edge with two loops. (2) Otherwise, if every component is an irreflexive caterpillar or a partially bristled reflexive path, then approximately counting retractions to H is equivalent to approximately counting the independent sets of a bipartite graph—a problem that is complete in the approximate counting complexity class RH Π 1. (3) Finally, if none of these hold, then approximately counting retractions to H is equivalent to approximately counting the satisfying assignments of a Boolean formula. 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

The complexity of approximately counting retractions to square-free graphs

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 #BIS. As retractions generalise homomorphisms, our easiness results extend to the important problem of approximately counting homomorphisms. By giving new #BIS-easiness results, we now settle the complexity of approximately counting homomorphisms for a whole class of non-trivial graphs that were previously unresolved