A decomposition method for valued CSPs (original) (raw)

A submodular-based decomposition strategy for valued CSPs

Valued Constraint Satisfaction Problems (VCSPs) can model many com-binatorial problems. VCSPs that involve submodular valuation functions only is a particular class of VCSPs that have the advantage of being tractable. In this pa-per, we propose a problem decomposition strategy for binary VCSPs which con-sists in decomposing the problem to be solved into a set of submodular, and then tractable, subproblems. The decomposition strategy combines two problem solving techniques, namely domain partitioning and value permutation.

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.

Submodularity-Based Decomposing for Valued CSP

International Journal on Artificial Intelligence Tools, 2013

Many combinatorial problems can be formulated as Valued Constraint Satisfaction Problems (VCSPs). In this framework, the constraints are defined by means of valuation functions to reflect several degrees of coherence. Despite the NP-hardness of the VCSP, tractable versions can be obtained by forcing the allowable valuation functions to have specific features. This is the case for submodular VCSPs, i.e. VCSPs that involve submodular valuation functions only. In this paper, we propose a problem decomposition scheme for binary VCSPs that takes advantage of submodular functions even when the studied problem is not submodular. The proposed scheme consists in decomposing the problem to be solved into a set of submodular, then tractable, subproblems. The decomposition scheme combines two techniques that where already used in the framework of constraint-based reasoning, but in separate manner. These techniques are domain partitioning and value permutation.

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.

Domain Reduction for Valued Constraints by Generalising Methods from CSP

Principles and Practice of Constraint Programming, 2018

For classical CSPs, the absence of broken triangles on a pair of values allows the merging of these values without changing the satisfiability of the instance, giving experimentally verified reduction in search time. We generalise the notion of broken triangles to VCSPs to obtain a tractable value-merging rule which preserves the cost of a solution. We then strengthen this value merging rule by using soft arc consistency to remove soft broken triangles and we show that the combined rule generalises known notions of domain value substitutability and interchangeability. Unfortunately the combined rules are no longer tractable to apply, but can still have applications as heuristics for reducing the search space. Finally we consider the generalisation of another value-elimination rule for CSPs to binary VCSPs. This new rule properly generalises neighbourhood substitutability and so we expect it will also have practical applications.

Binarisation for Valued Constraint Satisfaction Problems

SIAM Journal on Discrete Mathematics, 2017

We study methods for transforming valued constraint satisfaction problems (VCSPs) to binary VCSPs. First, we show that the standard dual encoding preserves many aspects of the algebraic properties that capture the computational complexity of VCSPs. Second, we extend the reduction of CSPs to binary CSPs described by Bulín et al. [LMCS'15] to VCSPs. This reduction establishes that VCSPs over a fixed valued constraint language are polynomial-time equivalent to Minimum-Cost Homomorphism Problems over a fixed digraph.

Solving weighted CSPs with meta-constraints by reformulation into satisfiability modulo theories

We introduce WSimply, a new framework for modelling and solving Weighted Constraint Satisfaction Problems (WCSP) using Satisfiability Modulo Theories (SMT) technology. In contrast to other well-known approaches designed for extensional representation of goods or no-goods, and with few declarative facilities, our approach aims to follow an intensional and declarative syntax style. In addition, our language has built-in support for some meta-constraints, such as priority and homogeneity, which allows the user to easily specify rich requirements on the desired solutions, such as preferences and fairness. We propose two alternative strategies for solving these WCSP instances using SMT. The first is the reformulation into Weighted SMT (WSMT) and the application of satisfiability test based algorithms from recent contributions in the Weighted Maximum Satisfiability field. The second one is the reformulation into an operation research-like style which involves an This work is an extended version of the paper [1] presented at the ModRef 2011 workshop. Constraints (2013) 18:236-268 237 optimisation variable or objective function and the application of optimisation SMT solvers. We present experimental results of two well-known problems: the Nurse Rostering Problem (NRP) and a variant of the Balanced Academic Curriculum Problem (BACP), and provide some insights into the impact of the addition of meta-constraints on the quality of the solutions and the solving time.

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.

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.

A Decomposition Technique for Solving {Max-CSP}

The objective of the Maximal Constraint Satisfaction Problem (Max-CSP) is to find an instantiation which minimizes the number of constraint violations in a constraint network. In this paper, inspired from the concept of inferred disjunctive constraints introduced by Freuder and Hubbe, we show that it is possible to exploit the arc-inconsistency counts, associated with each value of a network, in order to avoid exploring useless portions of the search space. The principle is to reason from the distance between the two best values in the domain of a variable, according to such counts. From this reasoning, we can build a decomposition technique which can be used throughout search in order to decompose the current problem into easier sub-problems. Interestingly, this approach does not depend on the structure of the constraint graph, as it is usually proposed. Alternatively, we can dynamically post hard constraints that can be used locally to prune the search space. The practical interest of our approach is illustrated, using this alternative, with an experimentation based on a classical branch and bound algorithm, namely PFC-MRDAC.