2,885 research outputs found
The Complexity of Counting Homomorphisms to Cactus Graphs Modulo 2
A homomorphism from a graph G to a graph H is a function from V(G) to V(H)
that preserves edges. Many combinatorial structures that arise in mathematics
and computer science can be represented naturally as graph homomorphisms and as
weighted sums of graph homomorphisms. In this paper, we study the complexity of
counting homomorphisms modulo 2. The complexity of modular counting was
introduced by Papadimitriou and Zachos and it has been pioneered by Valiant who
famously introduced a problem for which counting modulo 7 is easy but counting
modulo 2 is intractable. Modular counting provides a rich setting in which to
study the structure of homomorphism problems. In this case, the structure of
the graph H has a big influence on the complexity of the problem. Thus, our
approach is graph-theoretic. We give a complete solution for the class of
cactus graphs, which are connected graphs in which every edge belongs to at
most one cycle. Cactus graphs arise in many applications such as the modelling
of wireless sensor networks and the comparison of genomes. We show that, for
some cactus graphs H, counting homomorphisms to H modulo 2 can be done in
polynomial time. For every other fixed cactus graph H, the problem is complete
for the complexity class parity-P which is a wide complexity class to which
every problem in the polynomial hierarchy can be reduced (using randomised
reductions). Determining which H lead to tractable problems can be done in
polynomial time. Our result builds upon the work of Faben and Jerrum, who gave
a dichotomy for the case in which H is a tree.Comment: minor change
Counting Homomorphisms to Cactus Graphs Modulo 2
A homomorphism from a graph G to a graph H is a function from V(G) to V(H) that preserves edges. Many combinatorial structures that arise in mathematics and computer science can be represented naturally as graph homomorphisms and as weighted sums of graph homomorphisms. In this paper, we study the complexity of counting homomorphisms modulo 2. The complexity of modular counting was introduced by Papadimitriou and Zachos and it has been pioneered by Valiant who famously introduced a problem for which counting modulo 7 is easy but counting modulo 2 is intractable. Modular counting provides a rich setting in which to study the structure of homomorphism problems. In this case, the structure of the graph H has a big influence on the complexity of the problem. Thus, our approach is graph-theoretic. We give a complete solution for the class of cactus graphs, which are connected graphs in which every edge belongs to at most one cycle. Cactus graphs arise in many applications such as the modelling of wireless sensor networks and the comparison of genomes. We show that, for some cactus graphs H, counting homomorphisms to H modulo 2 can be done in polynomial time. For every other fixed cactus graph H, the problem is complete for the complexity class +P which is a wide complexity class to which every problem in the polynomial hierarchy can be reduced (using randomised reductions). Determining which H lead to tractable problems can be done in polynomial time. Our result builds upon the work of Faben and Jerrum, who gave a dichotomy for the case in which H is a tree
The Complexity of Approximately Counting Retractions
Let be a graph that contains an induced subgraph . A retraction from
to is a homomorphism from to that is the identity function on
. Retractions are very well-studied: Given , the complexity of deciding
whether there is a retraction from an input graph to is completely
classified, in the sense that it is known for which this problem is
tractable (assuming ). Similarly, the complexity of
(exactly) counting retractions from to is classified (assuming
). 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 . Our second contribution is to locate the retraction counting
problem for each 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
Counting Homomorphisms Modulo a Prime Number
Counting problems in general and counting graph homomorphisms in particular
have numerous applications in combinatorics, computer science, statistical
physics, and elsewhere. One of the most well studied problems in this area is
#GraphHom(H) --- the problem of finding the number of homomorphisms from a
given graph G to the graph H. Not only the complexity of this basic problem is
known, but also of its many variants for digraphs, more general relational
structures, graphs with weights, and others.
In this paper we consider a modification of #GraphHom(H), the #_p GraphHom(H)
problem, p a prime number: Given a graph G, find the number of homomorphisms
from G to H modulo p. In a series of papers Faben and Jerrum, and Goebel et al.
determined the complexity of #_2 GraphHom(H) in the case H (or, in fact, a
certain graph derived from H) is square-free, that is, does not contain a
4-cycle. Also, Goebel et al. found the complexity of #_p GraphHom(H) for an
arbitrary prime p when H is a tree. Here we extend the above result to show
that the #_p GraphHom(H) problem is #_p P-hard whenever the derived graph
associated with H is square-free and is not a star, which completely classifies
the complexity of #_p GraphHom(H) for square-free graphs H
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