The Complexity of Approximately Counting Tree Homomorphisms

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 J 3, 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 J 3 , 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 J 3 . 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 J 3. 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 ).