Finding solutions to a binary constraint satisfaction problem is known to be an NP-complete problem in general, but may be tractable in cases where either the set of allowed constraints or the graph structure is restricted. This paper considers restricted sets of contraints which are closed under permutation of the labels. We identify a set of constraints which gives rise to a class of tractable problems and give polynomial time algorithms for solving such problems, and for finding the equivalent minimal network. We also prove that the class of problems generated by any set of constraints not contained in this restricted set is NP-complete. 1 Introduction Finding solutions to a constraint satisfaction problem is known to be an NPcomplete problem in general [11] even when the constraints are restricted to binary constraints. However, many of the problems which arise in practice have special properties which allow them to be solved efficiently. The question of identifying restrictions t...

The Constraint Satisfaction Problem is a type of combinatorial search problem of much interest in Artificial Intelligence and Operations Research. The simplest algorithm for solving such a problem is chronological backtracking, but this method suffers from a malady known as "thrashing," in which essentially the same subproblems end up being solved repeatedly. Intelligent backtracking algorithms, such as backjumping and dependency-directed backtracking, were designed to address this difficulty, but the exact utility and range of applicability of these techniques have not been fully explored. This dissertation describes an experimental and theoretical investigation into the power of these intelligent backtracking algorithms. We compare the empirical performance of several such algorithms on a range of problem distributions. We show that the more sophisticated algorithms are especially useful on those problems with a small number of constraints that happen to be difficult for chronologica...

Constraint networks have beenshown to be useful in formulating such diverse problems as scene labeling, natural language parsing, and temporal reasoning. Given a constraint network, we often wish to (i) nd a solution that satis es the constraints and (ii) nd the corresponding minimal network where the constraints are as explicit as possible. Both tasks are known to be NP-complete in the general case. Task (i) is usually solved using a backtracking algorithm, and task (ii) is often solved only approximately by enforcing various levels of local consistency. In this paper, we identify a property of binary constraints called row convexity and show its usefulness in deciding when a form of local consistency called path consistency is sufficient to guarantee that a network is both minimal and globally consistent. Globally consistent networks have the property that a solution can be found without backtracking. We show that one can test for the row convexity property e ciently and we show, by examining

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This paper studies constraint satisfaction over connected row-convex (CRC) constraints. It shows that CRC constraints are closed under composition, intersection, and transposition, the basic operations of path-consistency algorithms. This establishes that path consistency over CRC constraints produces a minimal and decomposable network and is thus a polynomial-time decision procedure for CRC networks. This paper also presents a new path-consistency algorithm for CRC constraints running in time O(n3d2) and space O(n2d), wherenisthenumber of variables and d is the size of the largest domain, improving the traditional time and space complexity by orders of magnitude. The paper also shows how to construct CRC constraints by conjunction and disjunction of a set of basic CRC constraints, highlighting how CRC constraints generalize monotone constraints and presenting interesting subclasses of CRC constraints. Experimental results show that the algorithm behaves well in practice. © 1999 Elsevier Science B.V. All rights reserved.

We provide here a simple, yet very general framework that allows us to explain several constraint propagation algorithms in a systematic way. In particular, using the notions commutativity and semi-commutativity, we show how the well-known AC-3, PC-2, DAC and DPC algorithms are instances of a single generic algorithm. The work reported here extends and simplifies that of Apt [1].

This paper introduces a new level of partial consistency for constraint satisfaction problems, which is situated between arc and path-consistency. We call this level restricted path-consistency (rpc). Two procedures to enforce complete and partial rpc are presented. They both use path-based verifications to detect inconsistencies but retain the good features of arcconsistency since they only touch the variables domains and do not augment the connectivity of the constraint graph. We show that, although they perform a limited number of checks, these procedures exhibit a considerable pruning power. 1 Introduction Constraint satisfaction problems (csp) have proved useful to encode various instances of combinatorial problems. A csp is simply defined by giving a set of variables, each having a finite domain, and a set of constraints, each connected to a subset of the variables. Constraints are partial informations that restrict the values that can be assigned simultaneously to their variabl...

Dynamic Flexible Constraint Satisfaction and its Application to AI Planning Constraint satisfaction is a fundamental Arti cial Intelligence technique for knowledge representation and inference. It has, however, become clear that the original formulation of a static constraint satisfaction problem (CSP) with hard, imperative constraints is insucient to model many real problems. Recent work has addressed these shortcomings in the form of two separate extensions known as dynamic CSP and exible CSP respectively. Little has yet been done to combine dynamic and exible CSP in order to bring to bear the bene ts of both in solving more complex problems.

Abstract: Despite successful application of constraint programming (CP) to solving many real-life problems there is still an indispensable group or researchers considering (wrongly) CP as a simple evaluation technique only. Even if sophisticated search algorithms play an important role in solving constraint-based models, the real power engine behind CP is called constraint propagation (domain filtering, pruning or consistency techniques). In the paper we give a survey of common consistency techniques for binary constraints. We describe the main ideas behind them, list their advantages and limitations, and compare their pruning power. Then we briefly explain how these techniques can be extended to non-binary constraints. Last part of the paper is devoted to modelling issues. We give some hints how the constraint propagation can be exploited more when solving real-life problems. This part is based on our experience with solving real-life programs and it is also supported by empirical observations of other researchers.

One of the main factors limiting the use of path consistency algorithms in real life applications is their high space complexity. Han and Lee [5] presented a path consistency algorithm, PC-4, with O(n 3 a 3 ) space complexity, which makes it practicable only for small problems. I present a new path consistency algorithm, PC-5, which has an O(n 3 a 2 ) space complexity while retaining the worst-case time complexity of PC-4. Moreover, the new algorithm exhibits a much better average--case time complexity. The new algorithm is based on the idea (due to Bessiere [1]) that, at any time, only a minimal amount of support has to be found and recorded for a labeling to establish its viability; one has to look for a new support only if the current support is eliminated. I also show that PC-5 can be improved further to yield an algorithm, PC5++, with even better average-case performance and the same space complexity. 1 Introduction A large number of problems in AI can be posed as specia...