2,330 research outputs found
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
Approximation Complexity of Complex-Weighted Degree-Two Counting Constraint Satisfaction Problems
Constraint satisfaction problems have been studied in numerous fields with
practical and theoretical interests. In recent years, major breakthroughs have
been made in a study of counting constraint satisfaction problems (or #CSPs).
In particular, a computational complexity classification of bounded-degree
#CSPs has been discovered for all degrees except for two, where the "degree" of
an input instance is the maximal number of times that each input variable
appears in a given set of constraints. Despite the efforts of recent studies,
however, a complexity classification of degree-2 #CSPs has eluded from our
understandings. This paper challenges this open problem and gives its partial
solution by applying two novel proof techniques--T_{2}-constructibility and
parametrized symmetrization--which are specifically designed to handle
"arbitrary" constraints under randomized approximation-preserving reductions.
We partition entire constraints into four sets and we classify the
approximation complexity of all degree-2 #CSPs whose constraints are drawn from
two of the four sets into two categories: problems computable in
polynomial-time or problems that are at least as hard as #SAT. Our proof
exploits a close relationship between complex-weighted degree-2 #CSPs and
Holant problems, which are a natural generalization of complex-weighted #CSPs.Comment: A4, 10pt, 23 pages. This is a complete version of the paper that
appeared in the Proceedings of the 17th Annual International Computing and
Combinatorics Conference (COCOON 2011), Lecture Notes in Computer Science,
vol.6842, pp.122-133, Dallas, Texas, USA, August 14-16, 201
An event-based architecture for solving constraint satisfaction problems
Constraint satisfaction problems (CSPs) are typically solved using
conventional von Neumann computing architectures. However, these architectures
do not reflect the distributed nature of many of these problems and are thus
ill-suited to solving them. In this paper we present a hybrid analog/digital
hardware architecture specifically designed to solve such problems. We cast
CSPs as networks of stereotyped multi-stable oscillatory elements that
communicate using digital pulses, or events. The oscillatory elements are
implemented using analog non-stochastic circuits. The non-repeating phase
relations among the oscillatory elements drive the exploration of the solution
space. We show that this hardware architecture can yield state-of-the-art
performance on a number of CSPs under reasonable assumptions on the
implementation. We present measurements from a prototype electronic chip to
demonstrate that a physical implementation of the proposed architecture is
robust to practical non-idealities and to validate the theory proposed.Comment: First two authors contributed equally to this wor
A Dichotomy Theorem for the Approximate Counting of Complex-Weighted Bounded-Degree Boolean CSPs
We determine the computational complexity of approximately counting the total
weight of variable assignments for every complex-weighted Boolean constraint
satisfaction problem (or CSP) with any number of additional unary (i.e., arity
1) constraints, particularly, when degrees of input instances are bounded from
above by a fixed constant. All degree-1 counting CSPs are obviously solvable in
polynomial time. When the instance's degree is more than two, we present a
dichotomy theorem that classifies all counting CSPs admitting free unary
constraints into exactly two categories. This classification theorem extends,
to complex-weighted problems, an earlier result on the approximation complexity
of unweighted counting Boolean CSPs of bounded degree. The framework of the
proof of our theorem is based on a theory of signature developed from Valiant's
holographic algorithms that can efficiently solve seemingly intractable
counting CSPs. Despite the use of arbitrary complex weight, our proof of the
classification theorem is rather elementary and intuitive due to an extensive
use of a novel notion of limited T-constructibility. For the remaining degree-2
problems, in contrast, they are as hard to approximate as Holant problems,
which are a generalization of counting CSPs.Comment: A4, 10pt, 20 pages. This revised version improves its preliminary
version published under a slightly different title in the Proceedings of the
4th International Conference on Combinatorial Optimization and Applications
(COCOA 2010), Lecture Notes in Computer Science, Springer, Vol.6508 (Part I),
pp.285--299, Kailua-Kona, Hawaii, USA, December 18--20, 201
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
Detecting and Explaining Conflicts in Attributed Feature Models
Product configuration systems are often based on a variability model. The
development of a variability model is a time consuming and error-prone process.
Considering the ongoing development of products, the variability model has to
be adapted frequently. These changes often lead to mistakes, such that some
products cannot be derived from the model anymore, that undesired products are
derivable or that there are contradictions in the variability model. In this
paper, we propose an approach to discover and to explain contradictions in
attributed feature models efficiently in order to assist the developer with the
correction of mistakes. We use extended feature models with attributes and
arithmetic constraints, translate them into a constraint satisfaction problem
and explore those for contradictions. When a contradiction is found, the
constraints are searched for a set of contradicting relations by the
QuickXplain algorithm.Comment: In Proceedings FMSPLE 2015, arXiv:1504.0301
Stochastic Constraint Programming
To model combinatorial decision problems involving uncertainty and
probability, we introduce stochastic constraint programming. Stochastic
constraint programs contain both decision variables (which we can set) and
stochastic variables (which follow a probability distribution). They combine
together the best features of traditional constraint satisfaction, stochastic
integer programming, and stochastic satisfiability. We give a semantics for
stochastic constraint programs, and propose a number of complete algorithms and
approximation procedures. Finally, we discuss a number of extensions of
stochastic constraint programming to relax various assumptions like the
independence between stochastic variables, and compare with other approaches
for decision making under uncertainty.Comment: Proceedings of the 15th Eureopean Conference on Artificial
Intelligenc
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