Affiliation:
1. University of Oxford
2. Queen Mary, University of London
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
).
Funder
European Research Council
Seventh Framework Programme
Engineering and Physical Sciences Research Council
Publisher
Association for Computing Machinery (ACM)
Subject
Computational Theory and Mathematics,Theoretical Computer Science
Cited by
6 articles.
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