2,807 research outputs found
Hybrid tractability of soft constraint problems
The constraint satisfaction problem (CSP) is a central generic problem in
computer science and artificial intelligence: it provides a common framework
for many theoretical problems as well as for many real-life applications. Soft
constraint problems are a generalisation of the CSP which allow the user to
model optimisation problems. Considerable effort has been made in identifying
properties which ensure tractability in such problems. In this work, we
initiate the study of hybrid tractability of soft constraint problems; that is,
properties which guarantee tractability of the given soft constraint problem,
but which do not depend only on the underlying structure of the instance (such
as being tree-structured) or only on the types of soft constraints in the
instance (such as submodularity). We present several novel hybrid classes of
soft constraint problems, which include a machine scheduling problem,
constraint problems of arbitrary arities with no overlapping nogoods, and the
SoftAllDiff constraint with arbitrary unary soft constraints. An important tool
in our investigation will be the notion of forbidden substructures.Comment: A full version of a CP'10 paper, 26 page
Hybrid VCSPs with crisp and conservative valued templates
A constraint satisfaction problem (CSP) is a problem of computing a
homomorphism between two relational
structures. Analyzing its complexity has been a very fruitful research
direction, especially for fixed template CSPs, denoted , in
which the right side structure is fixed and the left side
structure is unconstrained.
Recently, the hybrid setting, written ,
where both sides are restricted simultaneously, attracted some attention. It
assumes that is taken from a class of relational structures
that additionally is closed under inverse homomorphisms. The last
property allows to exploit algebraic tools that have been developed for fixed
template CSPs. The key concept that connects hybrid CSPs with fixed-template
CSPs is the so called "lifted language". Namely, this is a constraint language
that can be constructed from an input . The
tractability of that language for any input is a
necessary condition for the tractability of the hybrid problem.
In the first part we investigate templates for which the
latter condition is not only necessary, but also is sufficient. We call such
templates widely tractable. For this purpose, we construct from
a new finite relational structure and define
as a class of structures homomorphic to . We
prove that wide tractability is equivalent to the tractability of
. Our proof is based on the key observation
that is homomorphic to if and only if the core of
is preserved by a Siggers polymorphism. Analogous
result is shown for valued conservative CSPs.Comment: 21 pages. arXiv admin note: text overlap with arXiv:1504.0706
Tree Projections and Constraint Optimization Problems: Fixed-Parameter Tractability and Parallel Algorithms
Tree projections provide a unifying framework to deal with most structural
decomposition methods of constraint satisfaction problems (CSPs). Within this
framework, a CSP instance is decomposed into a number of sub-problems, called
views, whose solutions are either already available or can be computed
efficiently. The goal is to arrange portions of these views in a tree-like
structure, called tree projection, which determines an efficiently solvable CSP
instance equivalent to the original one. Deciding whether a tree projection
exists is NP-hard. Solution methods have therefore been proposed in the
literature that do not require a tree projection to be given, and that either
correctly decide whether the given CSP instance is satisfiable, or return that
a tree projection actually does not exist. These approaches had not been
generalized so far on CSP extensions for optimization problems, where the goal
is to compute a solution of maximum value/minimum cost. The paper fills the
gap, by exhibiting a fixed-parameter polynomial-time algorithm that either
disproves the existence of tree projections or computes an optimal solution,
with the parameter being the size of the expression of the objective function
to be optimized over all possible solutions (and not the size of the whole
constraint formula, used in related works). Tractability results are also
established for the problem of returning the best K solutions. Finally,
parallel algorithms for such optimization problems are proposed and analyzed.
Given that the classes of acyclic hypergraphs, hypergraphs of bounded
treewidth, and hypergraphs of bounded generalized hypertree width are all
covered as special cases of the tree projection framework, the results in this
paper directly apply to these classes. These classes are extensively considered
in the CSP setting, as well as in conjunctive database query evaluation and
optimization
The complexity of finite-valued CSPs
We study the computational complexity of exact minimisation of
rational-valued discrete functions. Let be a set of rational-valued
functions on a fixed finite domain; such a set is called a finite-valued
constraint language. The valued constraint satisfaction problem,
, is the problem of minimising a function given as
a sum of functions from . We establish a dichotomy theorem with respect
to exact solvability for all finite-valued constraint languages defined on
domains of arbitrary finite size.
We show that every constraint language either admits a binary
symmetric fractional polymorphism in which case the basic linear programming
relaxation solves any instance of exactly, or
satisfies a simple hardness condition that allows for a
polynomial-time reduction from Max-Cut to
Tractability in Constraint Satisfaction Problems: A Survey
International audienceEven though the Constraint Satisfaction Problem (CSP) is NP-complete, many tractable classes of CSP instances have been identified. After discussing different forms and uses of tractability, we describe some landmark tractable classes and survey recent theoretical results. Although we concentrate on the classical CSP, we also cover its important extensions to infinite domains and optimisation, as well as #CSP and QCSP
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