14 research outputs found
Weak Bases of Boolean Co-Clones
Universal algebra and clone theory have proven to be a useful tool in the
study of constraint satisfaction problems since the complexity, up to logspace
reductions, is determined by the set of polymorphisms of the constraint
language. For classifications where primitive positive definitions are
unsuitable, such as size-preserving reductions, weaker closure operations may
be necessary. In this article we consider strong partial clones which can be
seen as a more fine-grained framework than Post's lattice where each clone
splits into an interval of strong partial clones. We investigate these
intervals and give simple relational descriptions, weak bases, of the largest
elements. The weak bases have a highly regular form and are in many cases
easily relatable to the smallest members in the intervals, which suggests that
the lattice of strong partial clones is considerably simpler than the full
lattice of partial clones
An approximation trichotomy for Boolean #CSP
We give a trichotomy theorem for the complexity of approximately counting the
number of satisfying assignments of a Boolean CSP instance. Such problems are
parameterised by a constraint language specifying the relations that may be
used in constraints. If every relation in the constraint language is affine
then the number of satisfying assignments can be exactly counted in polynomial
time. Otherwise, if every relation in the constraint language is in the
co-clone IM_2 from Post's lattice, then the problem of counting satisfying
assignments is complete with respect to approximation-preserving reductions in
the complexity class #RH\Pi_1. This means that the problem of approximately
counting satisfying assignments of such a CSP instance is equivalent in
complexity to several other known counting problems, including the problem of
approximately counting the number of independent sets in a bipartite graph. For
every other fixed constraint language, the problem is complete for #P with
respect to approximation-preserving reductions, meaning that there is no fully
polynomial randomised approximation scheme for counting satisfying assignments
unless NP=RP
Do Hard SAT-Related Reasoning Tasks Become Easier in the Krom Fragment?
Many reasoning problems are based on the problem of satisfiability (SAT).
While SAT itself becomes easy when restricting the structure of the formulas in
a certain way, the situation is more opaque for more involved decision
problems. We consider here the CardMinSat problem which asks, given a
propositional formula and an atom , whether is true in some
cardinality-minimal model of . This problem is easy for the Horn
fragment, but, as we will show in this paper, remains -complete (and
thus -hard) for the Krom fragment (which is given by formulas in
CNF where clauses have at most two literals). We will make use of this fact to
study the complexity of reasoning tasks in belief revision and logic-based
abduction and show that, while in some cases the restriction to Krom formulas
leads to a decrease of complexity, in others it does not. We thus also consider
the CardMinSat problem with respect to additional restrictions to Krom formulas
towards a better understanding of the tractability frontier of such problems
Complexity Classifications for logic-based Argumentation
We consider logic-based argumentation in which an argument is a pair (Fi,al),
where the support Fi is a minimal consistent set of formulae taken from a given
knowledge base (usually denoted by De) that entails the claim al (a formula).
We study the complexity of three central problems in argumentation: the
existence of a support Fi ss De, the validity of a support and the relevance
problem (given psi is there a support Fi such that psi ss Fi?). When arguments
are given in the full language of propositional logic these problems are
computationally costly tasks, the validity problem is DP-complete, the others
are SigP2-complete. We study these problems in Schaefer's famous framework
where the considered propositional formulae are in generalized conjunctive
normal form. This means that formulae are conjunctions of constraints build
upon a fixed finite set of Boolean relations Ga (the constraint language). We
show that according to the properties of this language Ga, deciding whether
there exists a support for a claim in a given knowledge base is either
polynomial, NP-complete, coNP-complete or SigP2-complete. We present a
dichotomous classification, P or DP-complete, for the verification problem and
a trichotomous classification for the relevance problem into either polynomial,
NP-complete, or SigP2-complete. These last two classifications are obtained by
means of algebraic tools
Sparsification of SAT and CSP Problems via Tractable Extensions
Unlike polynomial kernelization in general, for which many non-trivial results and methods exist, only few non-trival algorithms are known for polynomial-time sparsification. Furthermore, excepting problems on restricted inputs (such as graph problems on planar graphs), most such results rely upon encoding the instance as a system of bounded-degree polynomial equations. In particular, for satisfiability (SAT) problems with a fixed constraint language Î, every previously known result is captured by this approach; for several such problems, this is known to be tight. In this work, we investigate the limits of this approachâin particular, does it really cover all cases of non-trivial polynomial-time sparsification? We generalize the method using tools from the algebraic approach to constraint satisfaction problems (CSP). Every constraint that can be modelled via a system of linear equations, over some finite field F, also admits a finite domain extension to a tractable CSP with a Maltsev polymorphism; using known algorithms for Maltsev languages, we can show that every problem of the latter type admits a âbasisâ of O(n) constraints, which implies a linear sparsification for the original problem. This generalization appears to be strict; other special cases include constraints modelled via group equations over some finite group G. For sparsifications of polynomial but super-linear size, we consider two extensions of this. Most directly, we can capture systems of bounded-degree polynomial equations in a âlift-and-projectâ manner, by finding Maltsev extensions for constraints over c-tuples of variables, for a basis with O(nc) constraints. Additionally, we may use extensions with k-edge polymorphisms instead of requiring a Maltsev polymorphism. We also investigate characterizations of when such extensions exist. We give an infinite sequence of partial polymorphisms Ï1, Ï2, âŠwhich characterizes whether a language Î has a Maltsev extension (of possibly infinite domain). In the complementary direction of proving lower bounds on kernelizability, we prove that for any language not preserved by Ï1, the corresponding SAT problem does not admit a kernel of size O(n2âΔ) for any Δ > 0 unless the polynomial hierarchy collapses
The expressive power of valued constraints: Hierarchies and collapses
In this paper we investigate the ways in which a fixed collection of valued constraints can be combined to express other valued constraints. We show that in some cases a large class of valued constraints, of all possible arities, can be expressed by using valued constraints of a fixed finite arity. We also show that some simple classes of valued constraints, including the set of all monotonie valued constraints with finite cost values, cannot be expressed by a subset of any fixed finite arity, and hence form an infinite hierarchy. © Springer-Verlag Berlin Heidelberg 2007
Connectivity of Boolean Satisfiability
For Boolean satisfiability problems, the structure of the solution space is
characterized by the solution graph, where the vertices are the solutions, and
two solutions are connected iff they differ in exactly one variable. For this
implicitly defined graph, we here study the st-connectivity and connectivity
problems.
Building on the work of Gopalan et al. ("The Connectivity of Boolean
Satisfiability: Computational and Structural Dichotomies", 2006/2009), we first
investigate satisfiability problems given by CSPs, more exactly CNF(S)-formulas
with constants (as considered in Schaefer's famous 1978 dichotomy theorem); we
prove a computational dichotomy for the st-connectivity problem, asserting that
it is either solvable in polynomial time or PSPACE-complete, and an aligned
structural dichotomy, asserting that the maximal diameter of connected
components is either linear in the number of variables, or can be exponential;
further, we show a trichotomy for the connectivity problem, asserting that it
is either in P, coNP-complete, or PSPACE-complete.
Next we investigate two important variants: CNF(S)-formulas without
constants, and partially quantified formulas; in both cases, we prove analogous
dichotomies for st-connectivity and the diameter; for for the connectivity
problem, we show a trichotomy in the case of quantified formulas, while in the
case of formulas without constants, we identify fragments of a possible
trichotomy.
Finally, we consider the connectivity issues for B-formulas, which are
arbitrarily nested formulas built from some fixed set B of connectives, and for
B-circuits, which are Boolean circuits where the gates are from some finite set
B; we prove a common dichotomy for both connectivity problems and the diameter;
for partially quantified B-formulas, we show an analogous dichotomy.Comment: PhD thesis, 82 pages, contains all results from the previous papers
arXiv:1312.4524, arXiv:1312.6679, and arXiv:1403.6165, plus additional
findings. arXiv admin note: text overlap with arXiv:cs/0609072 by other
author