Affiliation:
1. School of Computing Science, Simon Fraser University, Burnaby BC, Canada
2. Department of Computer Science, University of Oxford, Oxford, UK
Abstract
In this article, we study the complexity of counting Constraint Satisfaction Problems (CSPs) of the form #CSP(
C
, -), in which the goal is, given a relational structure
A
from a class
C
of structures and an arbitrary structure
B
, to find the number of homomorphisms from
A
to
B
. Flum and Grohe showed that #CSP(
C
, -) is solvable in polynomial time if
C
has bounded treewidth [FOCS’02]. Building on the work of Grohe [JACM’07] on decision CSPs, Dalmau and Jonsson then showed that if
C
is a recursively enumerable class of relational structures of bounded arity, then, assuming FPT≠ #W[1], there are no other cases of #CSP(
C
, -) solvable exactly in polynomial time (or even fixed-parameter time) [TCS’04].
We show that, assuming FPT ≠ W[1] (under randomised parameterised reductions) and for
C
satisfying certain general conditions, #CSP(
C
,-) is not solvable even
approximately
for
C
of unbounded treewidth; that is, there is no fixed parameter tractable (and thus also not fully polynomial) randomised approximation scheme for #CSP(
C
, -). In particular, our condition generalises the case when
C
is closed undertaking minors.
Funder
NSERC Discovery
Royal Society University Research Fellowship
European Union’s Horizon 2020 research and innovation programme
European Research Council
Publisher
Association for Computing Machinery (ACM)
Subject
Computational Theory and Mathematics,Theoretical Computer Science
Cited by
6 articles.
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