The complexity of soft constraint satisfaction (original) (raw)
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Soft Constraints: Complexity and Multimorphisms
Lecture Notes in Computer Science, 2003
Over the past few years there has been considerable progress in methods to systematically analyse the complexity of classical (crisp) constraint satisfaction problems with specified constraint types. One very powerful theoretical development in this area links the complexity of a set of classical constraints to a corresponding set of algebraic operations, known as polymorphisms. In this paper we begin a systematic investigation of the complexity of combinatorial optimisation problems expressed using various forms of soft constraints. We extend the notion of a polymorphism by introducing a more general algebraic operation, which we call a multimorphism. We show that a number of maximal tractable sets of soft constraints, both established and novel, can be characterised by the presence of particular multimorphisms.
J. Multiple Valued Log. Soft Comput., 2022
Constraint satisfaction problems (CSPs) are combinatorial problems with strong ties to universal algebra and clone theory. The recently proved CSP dichotomy theorem states that each finite-domain CSP is either solvable in polynomial time, or that it is NP-complete. However, among the intractable CSPs there is a seemingly large variance in how fast they can be solved by exponential-time algorithms, which cannot be explained by the classical algebraic approach based on polymorphisms. In this contribution we will survey an alternative approach based on partial polymorphisms, which is useful for studying the fine-grained complexity of NP-complete CSPs. Moreover, we will state and discuss some challenging open problems in this research field. 1 Algebraic Background We begin by providing a self-contained introduction to the underlying algebraic approach. The reader familiar with universal algebra and clone theory can safely skim the two following subsections. miguel.couceiro@{loria,Inria}...
The Complexity of Boolean Constraint Isomorphism
Lecture Notes in Computer Science, 2004
We consider the Boolean constraint isomorphism problem, that is, the problem of determining whether two sets of Boolean constraint applications can be made equivalent by renaming the variables. We show that depending on the set of allowed constraints, the problem is either coNP-hard and GI-hard, equivalent to graph isomorphism, or polynomial-time solvable. This establishes a complete classification of the complexity of the problem, and moreover, it identifies exactly all those cases in which Boolean constraint isomorphism is polynomial-time manyone equivalent to graph isomorphism, the best-known and best-examined isomorphism problem in theoretical computer science.
A Maximal Tractable Class of Soft Constraints
Journal of Artificial Intelligence Research, 2004
Many researchers in artificial intelligence are beginning to explore the use of soft constraints to express a set of (possibly conflicting) problem requirements. A soft constraint is a function defined on a collection of variables which associates some measure of desirability with each possible combination of values for those variables. However, the crucial question of the computational complexity of finding the optimal solution to a collection of soft constraints has so far received very little attention. In this paper we identify a class of soft binary constraints for which the problem of finding the optimal solution is tractable. In other words, we show that for any given set of such constraints, there exists a polynomial time algorithm to determine the assignment having the best overall combined measure of desirability. This tractable class includes many commonly-occurring soft constraints, such as 'as near as possible' or 'as soon as possible after', as well as ...
A new hybrid tractable class of soft constraint problems
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2010
The constraint satisfaction problem (CSP) is a central generic problem in artificial intelligence. Considerable effort has been made in identifying properties which ensure tractability in such problems. In this paper we study hybrid tractability of soft constraint problems; that is, properties which guarantee tractability of the given soft constraint problem, but properties 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 firstly present two hybrid classes of soft constraint problems defined by forbidden subgraphs in the structure of the instance. These classes allow certain combinations of binary crisp constraints together with arbitrary unary soft constraints. We then introduce the joint-winner property, which allows us to define a novel hybrid tractable class of soft constraint problems with soft binary and unary constraints. This class generalises the SoftAllDiff constraint with arbitrary unary soft constraints. We show that the jointwinner property is easily recognisable in polynomial time and present a polynomial-time algorithm based on maximum-flows for the class of soft constraint problems satisfying the joint-winner property. Moreover, we show that if cost functions can only take on two distinct values then this class is maximal.
06401 Executive Summary--Complexity of Constraints}
2006
In a constraint satisfaction problem, the goal is to find an assignment of values to a given set of variables so that certain specified constraints are satisfied. Constraint satisfaction problems were introduced in the 1970s to model computational problems encountered in scene processing. It was quickly realized, however, that constraint satisfaction gives rise to a powerful general framework in which a wide variety of combinatorial problems can be expressed. As a matter of fact, it has been asserted that "Constraint satisfaction has a unitary theoretical model with myriad practical applications" (A. Mackworth, Foreword to Constraint Processing by Rina Dechter, 2003). Indeed, in artificial intelligence and other areas of computer science the constraint satisfaction framework is widely regarded as a versatile and efficient way of modeling and solving a number of real-world problems, such as planning and scheduling, frequency assignment problems, image processing, programming language analysis, and natural language understanding. In database theory, it has been shown that the key problem of conjunctive-query evaluation is actually equivalent to the constraint satisfaction problem. Furthermore, certain central problems in combinatorial optimization can be represented as constraint satisfaction problems. Thus, nowadays constraint satisfaction problems (CSPs) are ubiquitous in many different areas of computer science, from artificial intelligence and database systems to circuit design, network optimization, and theory of programming languages. Consequently, it is important to analyze and pinpoint the computational complexity of certain algorithmic tasks related to constraint satisfaction. These include determining if a CSP has a solution (and, if so, finding such a solution), counting the number of solutions of a CSP, enumerating all solutions of a CSP, and finding the biggest number of constraints that can be simultaneously satisfied, if a CSP is unsatisfiable. Complexity-theoretic results about these tasks may have direct impact on, for instance, the design and processing of database query languages, or strategies in data-mining, or the design and implementation of planners. During the past two decades, an impressive array of diverse techniques from mathematical fields, such as propositional logic, model theory, Boolean function theory, universal algebra and combinatorics, have been used to analyze the computational complexity of algorithmic tasks related to CSPs. Although significant progress has been made on several fronts, some of the central questions remain unsolved so far; perhaps the most prominent of these is to obtain a complete classification of the complexity of CSPs over an arbitrary, but fixed, finite domain. One of the main aims of the Dagstuhl Seminar on Complexity of Constraints was to bring together researchers from all areas of activity in constraint satisfaction, so that they can communicate state-of-theart advances and embark on a systematic interaction that will enhance the synergy between the different areas.
On the Scope of the Universal-Algebraic Approach to Constraint Satisfaction
Logical Methods in Computer Science, 2012
The universal-algebraic approach has proved a powerful tool in the study of the computational complexity of constraint satisfaction problems (CSPs). This approach has previously been applied to the study of CSPs with finite or (infinite) ω-categorical templates. Our first result is an exact characterization of those CSPs that can be formulated with (a finite or) an ω-categorical template.
New Algebraic Tools for Constraint Satisfaction
2006
The Galois connection involving polymorphisms and coclones has received a lot of attention in regard to constraint satisfaction problems. However, it fails if we are interested in a reduction giving equivalence instead of only satisfiability-equivalence. We show how a similar Galois connection involving weaker closure operators can be applied for these problems. As an example of the usefulness of our construction, we show how to obtain very short proofs of complexity classifications in this context.
Non-dichotomies in Constraint Satisfaction Complexity
Automata, Languages and Programming, 2008
We show that every computational decision problem is polynomialtime equivalent to a constraint satisfaction problem (CSP) with an infinite template. We also construct for every decision problem L an ω-categorical template Γ such that L reduces to CSP(Γ ) and CSP(Γ ) is in coNP L (i.e., the class coNP with an oracle for L). CSPs with ω-categorical templates are of special interest, because the universal-algebraic approach can be applied to study their computational complexity. Furthermore, we prove that there are ω-categorical templates with coNP-complete CSPs and ω-categorical templates with coNP-intermediate CSPs, i.e., problems in coNP that are neither coNP-complete nor in P (unless P=coNP). To construct the coNP-intermediate CSP with ω-categorical template we modify the proof of Ladner's theorem. A similar modification allows us to also prove a non-dichotomy result for a class of left-hand side restricted CSPs, which was left open in . We finally show that if the so-called local-global conjecture for infinite constraint languages (over a finite domain) is false, then there is no dichotomy for the constraint satisfaction problem for infinite constraint languages.