The Goldilocks problem

2003 ◽  
Vol 17 (1) ◽  
pp. 3-11 ◽  
Author(s):  
TUDOR HULUBEI ◽  
EUGENE C. FREUDER ◽  
RICHARD J. WALLACE
Keyword(s):  
Constraint Satisfaction ◽  
Feasible Solution ◽  
Constraint Satisfaction Problem ◽  
Real Life ◽  
Constraint Satisfaction Problems ◽  
Major Focus ◽  
Huge Number ◽  
Arc Consistency ◽  
A Value ◽  
Practical Standpoint

Constraint-based reasoning is often used to represent and find solutions to configuration problems. In the field of constraint satisfaction, the major focus has been on finding solutions to difficult problems. However, many real-life configuration problems, although not extremely complicated, have a huge number of solutions, few of which are acceptable from a practical standpoint. In this paper we present a value ordering heuristic for constraint solving that attempts to guide search toward solutions that are acceptable. More specifically, by considering weights that are assigned to values and sets of values, the heuristic can guide search toward solutions for which the total weight is within an acceptable interval. Experiments with random constraint satisfaction problems demonstrate that, when a problem has numerous solutions, the heuristic makes search extremely efficient even when there are relatively few solutions that fall within the interval of acceptable weights. In these cases, an algorithm that is very effective for finding a feasible solution to a given constraint satisfaction problem (the “maintained arc consistency” algorithm or MAC) does not find a solution in the same weight interval within a reasonable time when it is run without the heuristic.

scholarly journals A fine-grained arc-consistency algorithm for non-normalized constraint satisfaction problems

2011 ◽  
Vol 21 (4) ◽  
pp. 733-744 ◽  
Author(s):  
Marlene Arangú ◽  
Miguel Salido
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Programming ◽  
Real Life ◽  
Search Space ◽  
Constraint Satisfaction Problems ◽  
Fine Grained ◽  
Arc Consistency ◽  
Life Problems ◽  
Programming Techniques ◽  
Np Complete

A fine-grained arc-consistency algorithm for non-normalized constraint satisfaction problems Constraint programming is a powerful software technology for solving numerous real-life problems. Many of these problems can be modeled as Constraint Satisfaction Problems (CSPs) and solved using constraint programming techniques. However, solving a CSP is NP-complete so filtering techniques to reduce the search space are still necessary. Arc-consistency algorithms are widely used to prune the search space. The concept of arc-consistency is bidirectional, i.e., it must be ensured in both directions of the constraint (direct and inverse constraints). Two of the most well-known and frequently used arc-consistency algorithms for filtering CSPs are AC3 and AC4. These algorithms repeatedly carry out revisions and require support checks for identifying and deleting all unsupported values from the domains. Nevertheless, many revisions are ineffective, i.e., they cannot delete any value and consume a lot of checks and time. In this paper, we present AC4-OP, an optimized version of AC4 that manages the binary and non-normalized constraints in only one direction, storing the inverse founded supports for their later evaluation. Thus, it reduces the propagation phase avoiding unnecessary or ineffective checking. The use of AC4-OP reduces the number of constraint checks by 50% while pruning the same search space as AC4. The evaluation section shows the improvement of AC4-OP over AC4, AC6 and AC7 in random and non-normalized instances.

PREPROCESSING QUANTIFIED CONSTRAINT SATISFACTION PROBLEMS WITH VALUE REORDERING AND DIRECTIONAL ARC AND PATH CONSISTENCY

2008 ◽  
Vol 17 (02) ◽  
pp. 321-337 ◽  
Author(s):  
KOSTAS STERGIOU
Keyword(s):  
Constraint Satisfaction ◽  
Early Stage ◽  
Constraint Satisfaction Problem ◽  
Constraint Satisfaction Problems ◽  
Combinatorial Problems ◽  
Arc Consistency ◽  
Solution Methods ◽  
Speed Up ◽  
Phase Transition Region ◽  
The Impact

The Quantified Constraint Satisfaction Problem (QCSP) is an extension of the CSP that can be used to model combinatorial problems containing contingency or uncertainty. It allows for universally quantified variables that can model uncertain actions and events, such as the unknown weather for a future party, or an opponent's next move in a game. Although interest in QCSPs is increasing in recent years, the development of techniques for handling QCSPs is still at an early stage. For example, although it is well known that local consistencies are of primary importance in CSPs, only arc consistency has been extended to quantified problems. In this paper we contribute towards the development of solution methods for QCSPs in two ways. First, by extending directional arc and path consistency, two popular local consistencies in constraint satisfaction, to the quantified case and proposing an algorithm that achieves these consistencies. Second, by showing how value ordering heuristics can be utilized to speed up computation in QCSPs. We study the impact of preprocessing QCSPs with value reordering and directional quantified arc and path consistency by running experiments on randomly generated problems. Results show that our preprocessing methods can significantly speed up the QCSP solving process, especially on hard instances from the phase transition region.

scholarly journals Early and Efficient Identification of Useless Constraint Propagation for Alldifferent Constraints

2020 ◽  
Author(s):  
Xizhe Zhang ◽  
Jian Gao ◽  
Yizhi Lv ◽  
Weixiong Zhang
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Propagation ◽  
Constraint Satisfaction Problems ◽  
Benchmark Problem ◽  
Value Assignment ◽  
Arc Consistency ◽  
A Value ◽  
Removable Edge ◽  
Problem Instances ◽  
State Of Art

Constraints propagation and backtracking are two basic techniques for solving constraint satisfaction problems (CSPs). During the search for a solution, the variable and value pairs that do not belong to any solution can be discarded by constraint propagation to ensure generalized arc consistency so as to avoid the fruitless search. However, constraint propagation is frequently invoked often with little effect on many CSPs. Much effort has been devoted to predicting when to invoke constraint propagation for solving a CSP; however, no effective approach has been developed for the alldifferent constraint. Here we present a novel theorem for identifying the edges in a value graph of alldifferent constraint whose removal can significantly reduce useless constraint propagation. We prove that if an alternating cycle exists for a prospectively removable edge that represents a variable-value assignment, the edge (and the assignment) can be discarded without constraint propagation. Based on this theorem, we developed a novel optimizing technique for early detection of useless constraint propagation which can be incorporated in any existing algorithm for alldifferent constraint. Our implementation of the new method achieved speedup by a factor of 1-5 over the state-of-art approaches on 93 benchmark problem instances in 8 domains. Furthermore, the new algorithm is scalable well and runs increasingly faster than the existing methods on larger problems.

scholarly journals Binary Encodings of Non-binary Constraint Satisfaction Problems: Algorithms and Experimental Results

2005 ◽  
Vol 24 ◽  
pp. 641-684 ◽  
Author(s):  
N. Samaras ◽  
K. Stergiou
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Constraint Satisfaction Problems ◽  
Binary Representation ◽  
Binary Constraints ◽  
Arc Consistency ◽  
Binary Problem ◽  
Binary Constraint ◽  
Binary Problems ◽  
Standard Techniques

A non-binary Constraint Satisfaction Problem (CSP) can be solved directly using extended versions of binary techniques. Alternatively, the non-binary problem can be translated into an equivalent binary one. In this case, it is generally accepted that the translated problem can be solved by applying well-established techniques for binary CSPs. In this paper we evaluate the applicability of the latter approach. We demonstrate that the use of standard techniques for binary CSPs in the encodings of non-binary problems is problematic and results in models that are very rarely competitive with the non-binary representation. To overcome this, we propose specialized arc consistency and search algorithms for binary encodings, and we evaluate them theoretically and empirically. We consider three binary representations; the hidden variable encoding, the dual encoding, and the double encoding. Theoretical and empirical results show that, for certain classes of non-binary constraints, binary encodings are a competitive option, and in many cases, a better one than the non-binary representation.

scholarly journals Constraint programming viewed as rule-based programming

2001 ◽  
Vol 1 (6) ◽  
pp. 713-750 ◽  
Author(s):  
KRZYSZTOF R. APT ◽  
ERIC MONFROY
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Programming ◽  
Constraint Satisfaction Problems ◽  
Rule Based ◽  
Local Consistency ◽  
Arc Consistency ◽  
Computation Process ◽  
Rule Based Approach ◽  
Rule Consistency

We study here a natural situation when constraint programming can be entirely reduced to rule-based programming. To this end we explain first how one can compute on constraint satisfaction problems using rules represented by simple first-order formulas. Then we consider constraint satisfaction problems that are based on predefined, explicitly given constraints. To solve them we first derive rules from these explicitly given constraints and limit the computation process to a repeated application of these rules, combined with labeling. We consider two types of rule here. The first type, that we call equality rules, leads to a new notion of local consistency, called rule consistency that turns out to be weaker than arc consistency for constraints of arbitrary arity (called hyper-arc consistency in Marriott & Stuckey (1998)). For Boolean constraints rule consistency coincides with the closure under the well-known propagation rules for Boolean constraints. The second type of rules, that we call membership rules, yields a rule-based characterization of arc consistency. To show feasibility of this rule-based approach to constraint programming, we show how both types of rules can be automatically generated, as CHR rules of Frühwirth (1995). This yields an implementation of this approach to programming by means of constraint logic programming. We illustrate the usefulness of this approach to constraint programming by discussing various examples, including Boolean constraints, two typical examples of many valued logics, constraints dealing with Waltz's language for describing polyhedral scenes, and Allen's qualitative approach to temporal logic.

On Considering Constraints of Different Importance in Fuzzy Constraint Satisfaction Problems

1998 ◽  
Vol 06 (05) ◽  
pp. 489-501 ◽  
Author(s):  
Vicenç Torra I Reventós
Keyword(s):  
Constraint Satisfaction ◽  
Real World ◽  
Constraint Satisfaction Problems ◽  
Human Reasoning ◽  
Fuzzy Constraints ◽  
A Value ◽  
Equal Importance ◽  
Real World Applications ◽  
Fuzzy Constraint ◽  
Fuzzy Constraint Satisfaction

Several real-world applications (e.g., scheduling, configuration, …) can be formulated as Constraint Satisfaction Problems (CSP). In these cases, a set of variables have to be settled to a value with the requirement that they satisfy a set of constraints. Classical CSPs are defined only by means of crisp (Boolean) constraints. However, as sometimes Boolean constraints are too strict in relation to human reasoning, fuzzy constraints were introduced. When fuzzy constraints are considered, human reasoning usually performs some compensation between alternatives. Thus other operators than t-norms are advisable. Besides of that, not all constraints can be considered with equal importance. In this paper we show that the WOWA operator can consider both aspects: compensation between constraints and constraints of different importance.

Computation of the Usage Contexts Coverage of a Jigsaw With CSP Techniques

2010 ◽  
Author(s):  
Bernard Yannou ◽  
Jiliang Wang ◽  
Pierre-Alain Yvars
Keyword(s):  
Constraint Satisfaction ◽  
Performance Prediction ◽  
Constraint Satisfaction Problem ◽  
Design Parameters ◽  
A Value ◽  
Product Service ◽  
Value Sets ◽  
Usage Context ◽  
A Performance

In the context of the Usage Context Based Design (UCBD) of a product-service, a taxonomy of variables is suggested to setup the link between the design parameters of a product-service and the part of a set of expected usages that may be covered. This paper implements a physics-based model to provide a performance prediction for each usage context that also depends on the user skill. The physics describing the behavior and consequently the performances of a jigsaw are established. Simulating numerically the usage coverage is non trivial for two reasons: the presence of circular references in physical relations and the need to efficiently propagate value sets or domains instead of accurate values. For these two reasons, we modeled the usage coverage issue as a Constraint Satisfaction Problem and we result in the expected service performances and a value of a covered usage indicator.

FACTORED ARC CONSISTENCY FOR RETE MATCH

1993 ◽  
Vol 02 (02) ◽  
pp. 277-306
Author(s):  
MARK W. PERLIN
Keyword(s):  
Working Memory ◽  
Constraint Satisfaction ◽  
Production Systems ◽  
Constraint Satisfaction Problems ◽  
Unified Framework ◽  
Call Graph ◽  
Quadratic Dependence ◽  
Arc Consistency ◽  
Common Framework ◽  
Graph Factorization

RETE match for production systems and Arc Consistency (AC) filtering are two efficient AI algorithms that are designed for particular constraint satisfaction problems (CSP). Interestingly, it is possible to integrate the two within a common framework, and provide RETE with the lookahead advantages of AC. Unfortunately, the resulting quadratic dependence of AC on working memory (WM) size precludes the application of AC to RETE, since RETE responds to incremental changes in WM. Recent results in AC graph factorization, however, reduce the quadratic dependence to just linear for certain classes of problems, including RETE. In this paper, we present RETE and AC within the unified framework of Call-Graph Caching (CGC) evaluation. We describe factored arc consistency (FAC) and its use in an integrated FAC/RETE algorithm that provides RETE match with AC lookahead. We discuss our implementations, promising initial empirical results, and explore FAC/RETE’s applicability conditions and extensions. We conclude that incorporating factored AC into RETE match is an interesting and potentially useful application of arc consistency methods to the RETE CSP.

scholarly journals Constraint Satisfaction Problems and Their Reduction to Directed Graphs

2021 ◽  
Author(s):  
Muhanda Stella Mbaka Muzalal
Keyword(s):  
Constraint Satisfaction ◽  
Cell Phone ◽  
Constraint Satisfaction Problem ◽  
Directed Graphs ◽  
Oriented Graph ◽  
Constraint Satisfaction Problems ◽  
Systems Of Equations ◽  
Relational Structures ◽  
Algorithmic Problems ◽  
Boolean Formulas

Constraint satisfaction problems present a general framework for studying a large class of algorithmic problems such as satisfaction of Boolean formulas, solving systems of equations over finite fields, graph colourings, as well as various applied problems in artificial intelligence (scheduling, allocation of cell phone frequencies, among others.) CSP (Constraint Satisfaction Problems) bring together graph theory, complexity theory and universal algebra. It is a well known result, due to Feder and Vardi, that any constraint satisfaction problem over a finite relational structure can be reduced to the homomorphism problem for a finite oriented graph. Until recently, it was not known whether this reduction preserves the type of the algorithm which solves the original constraint satisfaction problem, so that the same algorithm solves the corresponding digraph homomorphism problem. We look at how a recent construction due to Bulin, Deli´c, Jackson, and Niven can be used to show that the polynomial solvability of a constraint satisfaction problem using Datalog, a programming language which is a weaker version of Prolog, translates from arbitrary relational structures to digraphs.

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