Abstract
We study two computational problems, parameterised by a fixed tree H. #HOMSTO(H) is the problem of counting homomorphisms from an input graph G to H. #WHOMSTO(H) is the problem of counting weighted homomorphisms to H, given an input graph G and a weight function for each vertex v of G. Even though H is a tree, these problems turn out to be sufficiently rich to capture all of the known approximation behaviour in #P. We give a complete trichotomy for #WHOMSTO(H). If H is a star, then #WHOMSTO(H) is in FP. If H is not a star but it does not contain a certain induced subgraph J3, then #WHOMSTO(H) is equivalent under approximation-preserving (AP) reductions to #BIS, the problem of counting independent sets in a bipartite graph. This problem is complete for the class #RHΠ1 under AP-reductions. Finally, if H contains an induced J3, then #WHOMSTO(H) is equivalent under AP-reductions to #SAT, the problem of counting satisfying assignments to a CNF Boolean formula. Thus, #WHOMSTO(H) is complete for #P under AP-reductions. The results are similar for #HOMSTO(H) except that a rich structure emerges if H contains an induced J3. We show that there are trees H for which #HOMSTO(H) is #SAT-equivalent (disproving a plausible conjecture of Kelk). However, it is still not known whether #HOMSTO(H) is #SAT-hard for every tree H which contains an induced J3. It turns out that there is an interesting connection between these homomorphism-counting problems and the problem of approximating the partition function of the ferromagnetic Potts model. In particular, we show that for a family of graphs Jq, parameterised by a positive integer q, the problem #HOMSTO(Jq) is AP-interreducible with the problem of approximating the partition function of the q-state Potts model. It was not previously known that the Potts model had a homomorphism-counting interpretation. We use this connection to obtain some additional upper bounds for the approximation complexity of #HOMSTO(Jq).
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Index Terms
The Complexity of Approximately Counting Tree Homomorphisms
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