17,384 research outputs found
Tractable Optimization Problems through Hypergraph-Based Structural Restrictions
Several variants of the Constraint Satisfaction Problem have been proposed
and investigated in the literature for modelling those scenarios where
solutions are associated with some given costs. Within these frameworks
computing an optimal solution is an NP-hard problem in general; yet, when
restricted over classes of instances whose constraint interactions can be
modelled via (nearly-)acyclic graphs, this problem is known to be solvable in
polynomial time. In this paper, larger classes of tractable instances are
singled out, by discussing solution approaches based on exploiting hypergraph
acyclicity and, more generally, structural decomposition methods, such as
(hyper)tree decompositions
Structural Decompositions for Problems with Global Constraints
A wide range of problems can be modelled as constraint satisfaction problems
(CSPs), that is, a set of constraints that must be satisfied simultaneously.
Constraints can either be represented extensionally, by explicitly listing
allowed combinations of values, or implicitly, by special-purpose algorithms
provided by a solver.
Such implicitly represented constraints, known as global constraints, are
widely used; indeed, they are one of the key reasons for the success of
constraint programming in solving real-world problems. In recent years, a
variety of restrictions on the structure of CSP instances have been shown to
yield tractable classes of CSPs. However, most such restrictions fail to
guarantee tractability for CSPs with global constraints. We therefore study the
applicability of structural restrictions to instances with such constraints.
We show that when the number of solutions to a CSP instance is bounded in key
parts of the problem, structural restrictions can be used to derive new
tractable classes. Furthermore, we show that this result extends to
combinations of instances drawn from known tractable classes, as well as to CSP
instances where constraints assign costs to satisfying assignments.Comment: The final publication is available at Springer via
http://dx.doi.org/10.1007/s10601-015-9181-
On The Power of Tree Projections: Structural Tractability of Enumerating CSP Solutions
The problem of deciding whether CSP instances admit solutions has been deeply
studied in the literature, and several structural tractability results have
been derived so far. However, constraint satisfaction comes in practice as a
computation problem where the focus is either on finding one solution, or on
enumerating all solutions, possibly projected to some given set of output
variables. The paper investigates the structural tractability of the problem of
enumerating (possibly projected) solutions, where tractability means here
computable with polynomial delay (WPD), since in general exponentially many
solutions may be computed. A general framework based on the notion of tree
projection of hypergraphs is considered, which generalizes all known
decomposition methods. Tractability results have been obtained both for classes
of structures where output variables are part of their specification, and for
classes of structures where computability WPD must be ensured for any possible
set of output variables. These results are shown to be tight, by exhibiting
dichotomies for classes of structures having bounded arity and where the tree
decomposition method is considered
Tractable Combinations of Global Constraints
We study the complexity of constraint satisfaction problems involving global
constraints, i.e., special-purpose constraints provided by a solver and
represented implicitly by a parametrised algorithm. Such constraints are widely
used; indeed, they are one of the key reasons for the success of constraint
programming in solving real-world problems.
Previous work has focused on the development of efficient propagators for
individual constraints. In this paper, we identify a new tractable class of
constraint problems involving global constraints of unbounded arity. To do so,
we combine structural restrictions with the observation that some important
types of global constraint do not distinguish between large classes of
equivalent solutions.Comment: To appear in proceedings of CP'13, LNCS 8124. arXiv admin note: text
overlap with arXiv:1307.179
Beyond Hypertree Width: Decomposition Methods Without Decompositions
The general intractability of the constraint satisfaction problem has
motivated the study of restrictions on this problem that permit polynomial-time
solvability. One major line of work has focused on structural restrictions,
which arise from restricting the interaction among constraint scopes. In this
paper, we engage in a mathematical investigation of generalized hypertree
width, a structural measure that has up to recently eluded study. We obtain a
number of computational results, including a simple proof of the tractability
of CSP instances having bounded generalized hypertree width
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
Efficient Solving of Quantified Inequality Constraints over the Real Numbers
Let a quantified inequality constraint over the reals be a formula in the
first-order predicate language over the structure of the real numbers, where
the allowed predicate symbols are and . Solving such constraints is
an undecidable problem when allowing function symbols such or . In
the paper we give an algorithm that terminates with a solution for all, except
for very special, pathological inputs. We ensure the practical efficiency of
this algorithm by employing constraint programming techniques
Limits of Preprocessing
We present a first theoretical analysis of the power of polynomial-time
preprocessing for important combinatorial problems from various areas in AI. We
consider problems from Constraint Satisfaction, Global Constraints,
Satisfiability, Nonmonotonic and Bayesian Reasoning. We show that, subject to a
complexity theoretic assumption, none of the considered problems can be reduced
by polynomial-time preprocessing to a problem kernel whose size is polynomial
in a structural problem parameter of the input, such as induced width or
backdoor size. Our results provide a firm theoretical boundary for the
performance of polynomial-time preprocessing algorithms for the considered
problems.Comment: This is a slightly longer version of a paper that appeared in the
proceedings of AAAI 201
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