Maintaining Arc Consistency Research Articles

Conflict history based heuristic for constraint satisfaction problem solving

The variable ordering heuristic is an important module in algorithms dedicated to solve Constraint Satisfaction Problems (CSP), while it impacts the efficiency of exploring the search space and the size of the search tree. It also exploits, often implicitly, the structure of the instances. In this paper, we propose Conflict-History Search (CHS), a dynamic and adaptive variable ordering heuristic for CSP solving. It is based on the search failures and considers the temporality of these failures throughout the solving steps. The exponential recency weighted average is used to estimate the evolution of the hardness of constraints throughout the search. The experimental evaluation on XCSP3 instances shows that integrating CHS to solvers based on MAC (Maintaining Arc Consistency) and BTD (Backtracking with Tree Decomposition) achieves competitive results and improvements compared to the state-of-the-art heuristics. Beyond the decision problem, we show empirically that the solving of the constraint optimization problem (COP) can also take advantage of this heuristic.

Narrowing Support Searching Range in Maintaining Arc Consistency for Solving Constraint Satisfaction Problems

Arc consistency is the most popular filtering technique for solving constraint satisfaction problems. Constraint check plays a central role in establishing arc consistency. In this paper, we propose a method to save constraint checks in maintaining coarse-grained arc consistency during backtracking search for solving the constraint satisfaction problems. We reduce the support searching range by utilizing the information generated by an AC3.1 algorithm at preprocessing step. Compared with the existing maintaining arc consistency (MAC) algorithms, the proposed MAC3 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">be</sub> algorithm saves constraint checks without maintaining additional data structures at each search tree node. Our experimental results show that MAC3 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">be</sub> saves both constraint checks and CPU time while solving some benchmark problems.

Computing and restoring global inverse consistency in interactive constraint satisfaction

Some applications require the interactive resolution of a constraint problem by a human user. In such cases, it is highly desirable that the person who interactively solves the problem is not given the choice to select values that do not lead to solutions. We call this property global inverse consistency. Existing systems simulate this either by maintaining arc consistency after each assignment performed by the user or by compiling offline the problem as a multi-valued decision diagram. In this article, we define several questions related to global inverse consistency and analyze their complexity. Despite their theoretical intractability, we propose several algorithms for enforcing and restoring global inverse consistency and we show that the best version is efficient enough to be used in an interactive setting on several configuration and design problems.

A hybrid algorithm for the robust graph coloring problem

A hybridalgorithm which combines mathematical programming techniques (Kruskal’s algorithm and the strategy of maintaining arc consistency to solve constraint satisfaction problem “CSP”) and heuristic methods (musical composition method and DSATUR) to resolve the robust graph coloring problem (RGCP) is proposed in this paper. Experimental result shows that this algorithm is better than the other algorithms presented on the literature.

General game playing with stochastic CSP

The challenge of General Game Playing (GGP) is to devise game playing programs that take as input the rules of any strategic game, described in the Game Description Language (GDL), and that effectively play without human intervention. The aim of this paper is to address the GGP challenge by casting GDL games (potentially with chance events) into the Stochastic Constraint Satisfaction Problem (SCSP). The stochastic constraint network of a game is decomposed into a sequence of µSCSPs (also know as one-stage SCSP), each associated with a game round. Winning strategies are searched by coupling the MAC (Maintaining Arc Consistency) algorithm, used to solve each µSCSP in turn, together with the UCB (Upper Confidence Bound) policy for approximating the values of those strategies obtained by the last µSCSP in the sequence. Extensive experiments conducted on various GDL games with different deliberation times per round, demonstrate that the MAC-UCB algorithm significantly outperforms the state-of-the-art UCT (Upper Confidence bounds for Trees) algorithm.

Systematic vs. Non-Systematic Search for 3D Aircraft Conflict Resolution

A conflict is an event in which two or more aircraft experience a loss of minimum separation. In this paper, we formulate the problem of solving conflicts arising among several aircraft moving in a shared airspace as a Constraint Satisfaction Problem (CSP). The constraint satisfaction problem being NP-complete, the algorithms developed to solve it have been of two types: non-systematic and systematic search methods. In this paper, we have considered a breakout algorithm as an example of non-systematic search methods and a backtracking procedure that maintains Arc Consistency (MAC) as an example of systematic search methods. The performance of these algorithms was compared experimentally and the Breakout algorithm is shown to be clearly superior.

Conditional and composite temporal CSPs

Constraint Satisfaction Problems (CSPs) have been widely used to solve combinatorial problems. In order to deal with dynamic CSPs where the information regarding any possible change is known a priori and can thus be enumerated beforehand, conditional constraints and composite variables have been studied in the past decade. Indeed, these two concepts allow the addition of variables and their related constraints in a dynamic manner during the resolution process. More precisely, a conditional constraint restricts the participation of a variable in a feasible scenario while a composite variable allows us to express a disjunction of variables where only one will be added to the problem to solve. In order to deal with a wide variety of real life applications under temporal constraints, we present in this paper a unique temporal CSP framework including numeric and symbolic temporal information, conditional constraints and composite variables. We call this model, a Conditional and Composite Temporal CSP (or CCTCSP). To solve the CCTCSP we propose two methods respectively based on Stochastic Local Search (SLS) and constraint propagation. In order to assess the efficiency in time of the solving methods we propose, experimental tests have been conducted on randomly generated CCTCSPs. The results demonstrate the superiority of a variant of the Maintaining Arc Consistency (MAC) technique (that we call MAX+) over the other constraint propagation strategies, Forward Checking (FC) and its variants, for both consistent and inconsistent problems. It has also been shown that, in the case of consistent problems, MAC+ outperforms the SLS method Min Conflict Random Walk (MCRW) for highly constrained CCTCSPs while both methods have comparable time performance for under and middle constrained problems. MCRW is, however, the method of choice for highly constrained CCTCSPs if we decide to trade search time for the quality of the solution returned (number of solved constraints).

Asynchronous aggregation and consistency in distributed constraint satisfaction

Constraint Satisfaction Problems (CSP) have been very successful in problem-solving tasks ranging from resource allocation and scheduling to configuration and design. Increasingly, many of these tasks pose themselves in a distributed setting where variables and constraints are distributed among different agents. A variety of asynchronous search algorithms have been proposed for addressing this setting. We show how two techniques commonly used in centralized constraint satisfaction, value aggregation and maintaining arc consistency can be applied to increase efficiency in an asynchronous, distributed context as well, and report on experiments that quantify the gains.

Solving weighted CSP by maintaining arc consistency

Recently, a general definition of arc consistency (AC) for soft constraint frameworks has been proposed by T. Schiex [Proc. CP-2000, Singapore, 2000, pp. 411–424]. In this paper we specialize this definition to weighted CSP and introduce two O( ed 3) enforcing algorithms. Then, we refine the definition and introduce a stronger form of arc consistency (AC∗) along with two O( n 2 d 2+ ed 3) algorithms. As in the CSP case, an important application of AC is to combine it with search. We empirically demonstrate that a branch and bound algorithm that maintains either AC or AC∗ is a state-of-the-art general solver for weighted CSP. Our experiments cover binary Max-CSP and Max-SAT problems.

Solution Techniques for Constraint Satisfaction Problems

A wide range of problems in Artificial Intelligence can be represented as instances of the Constraint Satisfaction Problem (CSP), and then be solved using some of the existing techniques for solving CSPs. In this paper, we start by defining the concept of CSP and showing how some combinatorial problems can be modelled as CSPs. Next, we give a detailed description of the basic techniques for constraint satisfaction: constraint propagation algorithms (node consistency, arc consistency, and kconsistency), search algorithms (generate and test, backtracking, backjumping, and conflict-directed backjumping), and hybrid algorithms (forward checking, and maintaining arc consistency).

Binary vs. non-binary constraints

There are two well known transformations from non-binary constraints to binary constraints applicable to constraint satisfaction problems (CSPs) with finite domains: the dual transformation and the hidden (variable) transformation. We perform a detailed formal comparison of these two transformations. Our comparison focuses on two backtracking algorithms that maintain a local consistency property at each node in their search tree: the forward checking and maintaining arc consistency algorithms. We first compare local consistency techniques such as arc consistency in terms of their inferential power when they are applied to the original (non-binary) formulation and to each of its binary transformations. For example, we prove that enforcing arc consistency on the original formulation is equivalent to enforcing it on the hidden transformation. We then extend these results to the two backtracking algorithms. We are able to give either a theoretical bound on how much one formulation is better than another, or examples that show such a bound does not exist. For example, we prove that the performance of the forward checking algorithm applied to the hidden transformation of a problem is within a polynomial bound of the performance of the same algorithm applied to the dual transformation of the problem. Our results can be used to help decide if applying one of these transformations to all (or part) of a constraint satisfaction model would be beneficial.

Determining if (FC-) (conflict-directed) backjumping visits a given node is NP-hard

Conflict-directed backjumping is a modification of the backtracking algorithm that can outperform forward checking in non-pathological examples. We prove it is in general NP-hard to determine if backjumping or conflict-directed backjumping or their forward checking hybrids visit a given node of a search space. This shows that these algorithms are fundamentally more complex to analyze than backtracking and forward checking. We conclude by describing how similar results can be proved for versions of the Maintaining Arc Consistency algorithm.

Conflict-Directed Backjumping Revisited

In recent years, many improvements to backtracking algorithms for solving constraint satisfaction problems have been proposed. The techniques for improving backtracking algorithms can be conveniently classified as look-ahead schemes and look-back schemes. Unfortunately, look-ahead and look-back schemes are not entirely orthogonal as it has been observed empirically that the enhancement of look-ahead techniques is sometimes counterproductive to the effects of look-back techniques. In this paper, we focus on the relationship between the two most important look-ahead techniques---using a variable ordering heuristic and maintaining a level of local consistency during the backtracking search---and the look-back technique of conflict-directed backjumping (CBJ). We show that there exists a ``perfect'' dynamic variable ordering such that CBJ becomes redundant. We also show theoretically that as the level of local consistency that is maintained in the backtracking search is increased, the less that backjumping will be an improvement. Our theoretical results partially explain why a backtracking algorithm doing more in the look-ahead phase cannot benefit more from the backjumping look-back scheme. Finally, we show empirically that adding CBJ to a backtracking algorithm that maintains generalized arc consistency (GAC), an algorithm that we refer to as GAC-CBJ, can still provide orders of magnitude speedups. Our empirical results contrast with Bessiere and Regin's conclusion (1996) that CBJ is useless to an algorithm that maintains arc consistency.