380,879 research outputs found
Descriptive Complexity for Counting Complexity Classes
Descriptive Complexity has been very successful in characterizing complexity
classes of decision problems in terms of the properties definable in some
logics. However, descriptive complexity for counting complexity classes, such
as FP and #P, has not been systematically studied, and it is not as developed
as its decision counterpart. In this paper, we propose a framework based on
Weighted Logics to address this issue. Specifically, by focusing on the natural
numbers we obtain a logic called Quantitative Second Order Logics (QSO), and
show how some of its fragments can be used to capture fundamental counting
complexity classes such as FP, #P and FPSPACE, among others. We also use QSO to
define a hierarchy inside #P, identifying counting complexity classes with good
closure and approximation properties, and which admit natural complete
problems. Finally, we add recursion to QSO, and show how this extension
naturally captures lower counting complexity classes such as #L
Constraint Satisfaction with Counting Quantifiers
We initiate the study of constraint satisfaction problems (CSPs) in the
presence of counting quantifiers, which may be seen as variants of CSPs in the
mould of quantified CSPs (QCSPs). We show that a single counting quantifier
strictly between exists^1:=exists and exists^n:=forall (the domain being of
size n) already affords the maximal possible complexity of QCSPs (which have
both exists and forall), being Pspace-complete for a suitably chosen template.
Next, we focus on the complexity of subsets of counting quantifiers on clique
and cycle templates. For cycles we give a full trichotomy -- all such problems
are in L, NP-complete or Pspace-complete. For cliques we come close to a
similar trichotomy, but one case remains outstanding. Afterwards, we consider
the generalisation of CSPs in which we augment the extant quantifier
exists^1:=exists with the quantifier exists^j (j not 1). Such a CSP is already
NP-hard on non-bipartite graph templates. We explore the situation of this
generalised CSP on bipartite templates, giving various conditions for both
tractability and hardness -- culminating in a classification theorem for
general graphs. Finally, we use counting quantifiers to solve the complexity of
a concrete QCSP whose complexity was previously open
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
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