76 research outputs found
Low-level dichotomy for Quantified Constraint Satisfaction Problems
Building on a result of Larose and Tesson for constraint satisfaction
problems (CSP s), we uncover a dichotomy for the quantified constraint
satisfaction problem QCSP(B), where B is a finite structure that is a core.
Specifically, such problems are either in ALogtime or are L-hard. This involves
demonstrating that if CSP(B) is first-order expressible, and B is a core, then
QCSP(B) is in ALogtime.
We show that the class of B such that CSP(B) is first-order expressible
(indeed, trivially true) is a microcosm for all QCSPs. Specifically, for any B
there exists a C such that CSP(C) is trivially true, yet QCSP(B) and QCSP(C)
are equivalent under logspace reductions
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
QCSP on semicomplete digraphs
We study the (non-uniform) quantified constraint satisfaction
problem QCSP(H) as H ranges over semicomplete digraphs. We
obtain a complexity-theoretic trichotomy: QCSP(H) is either in P, is NP-complete or is Pspace-complete. The largest part of our work is the algebraic classification of precisely which semicompletes enjoy only essentially unary polymorphisms, which is combinatorially interesting in its own right
Quantified Constraints in Twenty Seventeen
I present a survey of recent advances in the algorithmic and computational complexity theory of non-Boolean Quantified Constraint Satisfaction Problems, incorporating some more modern research directions
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