Domain Reduction for Valued Constraints by Generalising Methods from CSP (original) (raw)
Related papers
Extending Broken Triangles and Enhanced Value-Merging
Lecture Notes in Computer Science, 2016
Broken triangles constitute an important concept not only for solving constraint satisfaction problems in polynomial time, but also for variable elimination or domain reduction by merging domain values. Specifically, for a given variable in a binary arc-consistent CSP, if no broken triangle occurs on any pair of values, then this variable can be eliminated while preserving satisfiability. More recently, it has been shown that even when this rule cannot be applied, it could be possible that for a given pair of values no broken triangle occurs. In this case, we can apply a domain-reduction operation which consists in merging these values while preserving satisfiability. In this paper we show that under certain conditions, and even if there are some broken triangles on a pair of values, these values can be merged without changing the satisfiability of the instance. This allows us to define a stronger merging operation and a new tractable class of binary CSP instances. We report experimental trials on benchmark instances.
Artificial Intelligence, 2016
A binary CSP instance satisfying the broken-triangle property (BTP) can be solved in polynomial time. Unfortunately, in practice, few instances satisfy the BTP. We show that a local version of the BTP allows the merging of domain values in arbitrary instances of binary CSP, thus providing a novel polynomialtime reduction operation. Extensive experimental trials on benchmark instances demonstrate a significant decrease in instance size for certain classes of problems. We show that BTP-merging can be generalised to instances with constraints of arbitrary arity and we investigate the theoretical relationship with resolution in SAT. A directional version of general-arity BTP-merging then allows us to extend the BTP tractable class previously defined only for binary CSP. We investigate the complexity of several related problems including the recognition problem for the general-arity BTP class when the variable order is unknown, finding an optimal order in which to apply BTP merges and detecting BTP-merges in the presence of global constraints such as AllDifferent.
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.
Eliminating redundancy in CSPs through merging and subsumption of domain values
ACM SIGAPP Applied Computing Review, 2013
Onto-substitutability has been shown to be intrinsic to how a domain value is considered redundant in Constraint Satisfaction Problems (CSPs). A value is onto-substitutable if any solution involving that value remains a solution when that value is replaced by some other value. We redefine onto-substitutability to accommodate binary relationships and study its implication. Joint interchangeability, an extension of onto-substitutability to its interchangeability counterpart, emerges as one of the results. We propose a new way of removing interchangeable values by constructing a new value as an intermediate step, as well as introduce virtual interchangeability, a local reasoning that leads to joint interchangeability and allows values to be merged together. Algorithms for removing onto-substitutable values are also proposed. 1
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.
Variable Elimination in Binary CSPs (Extended Abstract)
2020
We investigate rules which allow variable elimination in binary CSP (constraint satisfaction problem) instances while conserving satisfiability. We propose new rules and compare them, both theoretically and experimentally. We give optimised algorithms to apply these rules and show that each defines a novel tractable class. Using our variable-elimination rules in preprocessing allowed us to solve more benchmark problems than without.
High-Order Consistency in Valued Constraint Satisfaction
Constraints, 2005
k-consistency operations in constraint satisfaction problems (CSPs) render constraints more explicit by solving size-k subproblems and projecting the information thus obtained down to low-order constraints. We generalise this notion of k-consistency to valued constraint satisfaction problems (VCSPs) and show that it can be established in polynomial time when penalties lie in a discrete valuation structure. A generic definition of consistency is given which can be tailored to particular applications. As an example, a version of high-order consistency (face consistency) is presented which can be established in low-order polynomial time given certain restrictions on the valuation structure and the form of the constraint graph.
Generalizing constraint satisfaction on trees: Hybrid tractability and variable elimination
Artificial Intelligence, 2010
The Constraint Satisfaction Problem (CSP) is a central generic problem in artificial intelligence. Considerable progress has been made in identifying properties which ensure tractability in such problems, such as the property of being tree-structured. In this paper we introduce the broken-triangle property, which allows us to define a novel tractable class for this problem which significantly generalizes the class of problems with tree structure. We show that the broken-triangle property is conservative (i.e., it is preserved under domain reduction and hence under arc consistency operations) and that there is a polynomial-time algorithm to determine an ordering of the variables for which the brokentriangle property holds (or to determine that no such ordering exists). We also present a non-conservative extension of the broken-triangle property which is also sufficient to ensure tractability and can also be detected in polynomial time. We show that both the broken-triangle property and its extension can be used to eliminate variables, and that both of these properties provide the basis for preprocessing procedures that yield unique closures orthogonal to value elimination by enforcement of consistency. Finally, we also discuss the possibility of using the broken-triangle property in variableordering heuristics.
A filtering technique to achieve 2-consistency in constraint satisfaction problems
2012
Arc-Consistency algorithms are the most commonly used filtering techniques to prune the search space in Constraint Satisfaction Problems (CSPs). 2-consistency is a similar technique that guarantees that any instantiation of a value to a variable can be consistently extended to any second variable. Thus, 2-consistency can be stronger than arc-consistency in binary CSPs. In this work we present a new algorithm to achieve 2consistency called 2-C4. This algorithm is a reformulation of AC4 algorithm that is able to reduce unnecessary checking and prune more search space than AC4. The experimental results show that 2-C4 was able to prune more search space than arc-consistency algorithms in non-normalized instances. Furthermore, 2-C4 was more efficient than other 2-consistency algorithms presented in the literature.