Binarisation for Valued Constraint Satisfaction Problems (original) (raw)
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Binarisation via Dualisation for Valued Constraints
Proceedings of the AAAI Conference on Artificial Intelligence
Constraint programming is a natural paradigm for many combinatorial optimisation problems. The complexity of constraint satisfaction for various forms of constraints has been widely-studied, both to inform the choice of appropriate algorithms, and to understand better the boundary between polynomial-time complexity and NP-hardness. In constraint programming it is well-known that any constraint satisfaction problem can be converted to an equivalent binary problem using the so-called dual encoding. Using this standard approach any fixed collection of constraints, of arbitrary arity, can be converted to an equivalent set of constraints of arity at most two. Here we show that this transformation, although it changes the domain of the constraints, preserves all the relevant algebraic properties that determine the complexity. Moreover, we show that the dual encoding preserves many of the key algorithmic properties of the original instance. We also show that this remains true for more gene...
Valued Constraint Satisfaction Problems: Hard and Easy Problems
1995
In order to deal with over-constrained Constraint Satisfaction Problems, various extensions of the CSP framework have been considered by taking into account costs, uncertainties, preferences, priorities...Each extension uses a specific mathematical operator (+, max...) to aggregate constraint violations. In this paper, we consider a simple algebraic framework, related to Partial Constraint Satisfaction, which subsumes most of these proposals and use it to characterize existing proposals in terms of rationality and computational complexity. We exhibit simple relationships between these proposals, try to extend some traditional CSP algorithms and prove that some of these extensions may be computationally expensive.
An algebraic theory of complexity for valued constraints: Establishing a Galois connection
Mathematical Foundations of Computer Science 2011, 2011
The complexity of any optimisation problem depends critically on the form of the objective function. Valued constraint satisfaction problems are discrete optimisation problems where the function to be minimised is given as a sum of cost functions defined on specified subsets of variables. These cost functions are chosen from some fixed set of available cost functions, known as a valued constraint language. We show in this paper that when the costs are non-negative rational numbers or infinite, then the complexity of a valued constraint problem is determined by certain algebraic properties of this valued constraint language, which we call weighted polymorphisms. We define a Galois connection between valued constraint languages and sets of weighted polymorphisms and show how the closed sets of this Galois connection can be characterised. These results provide a new approach in the search for tractable valued constraint languages.
A Study of Encodings of Constraint Satisfaction Problems with 0/1 Variables
CoLogNet Publications, 2002
Many constraint satisfaction problems (csp's) are formulated with 0/1 variables. Sometimes this is a natural encoding, sometimes it is as a result of a reformulation of the problem, other times 0/1 variables make up only a part of the problem. Frequently we have constraints that restrict the sum of the values of variables. This can be encoded as a simple summation of the variables. However, since variables can only take 0/1 values we can also use an occurrence constraint, e.g. the number of occurrences of 1 must be k. Would this make a difference? Similarly, problems may use channelling constraints and encode these as a biconditional such as P ↔ Q (i.e. P if and only if Q). This can also be encoded in a number of ways. Might this make a difference as well? We attempt to answer these questions, using a variety of problems and two constraint programming toolkits. We show that even minor changes to the formulation of a constraint can have a profound effect on the run time of a constraint program and that these effects are not consistent across constraint programming toolkits. This leads us to a cautionary note for constraint programmers: take note of how you encode constraints, and don't assume computational behaviour is toolkit independent.
Cyclic consistency: A local reduction operation for binary valued constraints
Artificial Intelligence, 2004
Valued constraint satisfaction provides a general framework for optimisation problems over finite domains. It is a generalisation of crisp constraint satisfaction allowing the user to express preferences between solutions. Consistency is undoubtedly the most important tool for solving crisp constraints. It is not only a family of simplification operations on problem instances; it also lies at the heart of intelligent search techniques [G. Kondrak, P. van Beek, Artificial Intelligence 89 (1997) 365-387] and provides the key to solving certain classes of tractable constraints [P.G. Jeavons, D.A. Cohen, M.C. Cooper, Artificial Intelligence 101 (1998) 251-265]. Arc consistency was generalised to valued constraints by sacrificing the uniqueness of the arc consistency closure [M.C. Cooper, T. Schiex, Artificial Intelligence, in press]. The notion of 3-cyclic consistency, introduced in this paper, again sacrifices the unique-closure property in order to obtain a generalisation of path consistency to valued constraints which is checkable in polynomial time. In MAX-CSP, 3-cyclic consistency can be established in polynomial time and even guarantees a local form of optimality. The space complexity of 3-cyclic consistency is optimal since it creates no new constraints.
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}...
On the encoding of constraint satisfaction problems with 0/1 variables
2001
Abstract. Many constraint satisfaction problems (csp's) are formulated with 0/1 variables. Sometimes this is a natural encoding, sometimes it is as a result of a reformulation of the problem, other times 0/1 variables make up only a part of the problem. Frequently we have constraints that restrict the sum of the values of variables. This can be encoded as a simple summation of the variables. However, since variables can only take 0/1 values we can also use an occurrence constraint, eg the number of occurrences of 1 must be k.
Domain permutation reduction for Valued CSPs
2011
Several combinatorial problems can be formulated as Valued Constraint Satisfaction Problems (VCSPs) where constraints are defined through the use of valuation functions to reflect degrees of coherence. The goal is to find an assignment of values to variables with an overall finite and optimal valuation. Despite the NP-hardness of this task, tractable versions can be obtained by forcing the allowable valuation functions to have specific features. This is the case, for instance, of VCSPs with binary and submodular valuation functions . In this paper, we are concerned with a problem generalizing submodular binary VCSP, which we will call permuted submodular binary VCSP. The latter problem is obtained by independently applying permutations on the domains of submodular binary VCSP. We show that VCSP instances built from permuted submodular binary functions satisfying an extra condition can be identified in O(n 2 d 4 ) steps and solved, by means of the algorithm used for submodular binary VCSPs [2], in O(n 3 d 3 ) steps, where n is the number of variables and d is the size of the largest domain.
Hybrid tractability of valued constraint problems
Artificial Intelligence, 2011
We introduce tractable classes of VCSP instances based on convex cost functions. Firstly, we show that the class of VCSP instances satisfying the hierarchically nested convexity property is tractable. This class generalises our recent results on VCSP instances satisfying the non-overlapping convexity property by dropping the assumption that the input functions are non-decreasing [3]. Not only do we generalise the tractable class from [3], but also our algorithm has better running time compared to the algorithm from [3]. We present several examples of applications including soft hierarchical global cardinality constraints, useful in rostering problems. We go on to show that, over Boolean domains, it is possible to determine in polynomial time whether there exists some subset of the constraints such that the VCSP satisfies the hierarchically nested convexity property after renaming the variables in these constraints. 1 Preliminaries VCSPs As usual, we denote by N the set of positive integers with zero, and by Q set of all rational numbers. We denote Q = Q ∪ {∞} with the standard addition operation extended so that for all α ∈ Q, α + ∞ = ∞. In a VCSP (Valued Constraint Satisfaction Problem) the objective function to be minimised is the sum of cost functions whose arguments are subsets of arbitrary size of the variables v 1 ,. .. , v n where the domain of v i is D i. For notational convenience, we interpret a solution x (i.e. an assignment to the variables v 1 ,. .. , v n) as the set of variable,value assignments { v i , x i : i = 1,. .. , n}. The range of all cost functions is Q. Network flows. Here we review some basics on flows in graphs. We refer the reader to the standard textbook [1] for more details. We present only the notions and results needed for our purposes. In particular, we deal with only integral flows. Let G = (V, A) be a directed graph with vertex set V and arc set A. To each arc a ∈ A we assign a demand/capacity function [d(a), c(a)] and a weight Martin Cooper is supported by ANR Projects ANR-10-BLAN-0210 and ANR-10-BLAN-0214. StanislavŽivný is supported by a Junior Research Fellowship at University College, Oxford.
A polynomial relational class of binary CSP
Annals of Mathematics and Artificial Intelligence, 2017
Finding a solution to a constraint satisfaction problem (CSP) is known to be an NP-hard task. Considerable effort has been spent on identifying tractable classes of CSP, in other words, classes of constraint satisfaction problems for which there are polynomial time recognition and resolution algorithms. In this article, we present a relational tractable class of binary CSP. Our key contribution is a new ternary operation that we name mjx. We first characterize mjx-closed relations which leads to an optimal algorithm to recognize such relations. To reduce space and time complexity, we define a new storage technique for these relations which reduces the complexity of establishing a form of strong directional path consistency, the consistency level that solves all instances of the proposed class (and, indeed, of all relational classes closed under a majority polymorphism). 1 Introduction Many real-world problems may be formulated by means of constraints on time, on space or more generally on resources. Planning, scheduling and resource allocation are just a few among many problems that involve reasoning about constraints. Such problems are designated by the general term constraint satisfaction problem (CSP) and are highly combinatorial, because their solutions are to be found among a huge set of combinations. In terms of complexity theory, solving a CSP is, in general, an NP-complete task. Nonetheless, many real word CSPs have specific properties that make them recognizable and solvable in polynomial time. Thus, despite the NP-completeness of the CSP, many of its instances fall into tractable