Generalised arc consistency for the AllDifferent constraint: An empirical survey

AbstractThe AllDifferent constraint is a crucial component of any constraint toolkit, language or solver, since it is very widely used in a variety of constraint models. The literature contains many different versions of this constraint, which trade strength of inference against computational cost. In this paper, we focus on the highest strength of inference, enforcing a property known as generalised arc consistency (GAC). This work is an analytical survey of optimizations of the main algorithm for GAC for the AllDifferent constraint. We evaluate empirically a number of key techniques from the literature. We also report important implementation details of those techniques, which have often not been described in published papers. We pay particular attention to improving incrementality by exploiting the strongly-connected components discovered during the standard propagation process, since this has not been detailed before. Our empirical work represents by far the most extensive set of experiments on variants of GAC algorithms for AllDifferent. Overall, the best combination of optimizations gives a mean speedup of 168 times over the same implementation without the optimizations

A Computational Model for Logical Analysis of Data

Initially introduced by Peter Hammer, Logical Analysis of Data is a methodology that aims at computing a logical justification for dividing a group of data in two groups of observations, usually called the positive and negative groups. Consider this partition into positive and negative groups as the description of a partially defined Boolean function; the data is then processed to identify a subset of attributes, whose values may be used to characterize the observations of the positive groups against those of the negative group. LAD constitutes an interesting rule-based learning alternative to classic statistical learning techniques and has many practical applications. Nevertheless, the computation of group characterization may be costly, depending on the properties of the data instances. A major aim of our work is to provide effective tools for speeding up the computations, by computing some \emph{a priori} probability that a given set of attributes does characterize the positive and negative groups. To this effect, we propose several models for representing the data set of observations, according to the information we have on it. These models, and the probabilities they allow us to compute, are also helpful for quickly assessing some properties of the real data at hand; furthermore they may help us to better analyze and understand the computational difficulties encountered by solving methods. Once our models have been established, the mathematical tools for computing probabilities come from Analytic Combinatorics. They allow us to express the desired probabilities as ratios of generating functions coefficients, which then provide a quick computation of their numerical values. A further, long-range goal of this paper is to show that the methods of Analytic Combinatorics can help in analyzing the performance of various algorithms in LAD and related fields

Revisiting Counting Solutions for the Global Cardinality Constraint

International audienceCounting solutions for a combinatorial problem has been identified as an important concern within the Artificial Intelligence field. It is indeed very helpful when exploring the structure of the solution space. In this context, this paper revisits the computation process to count solutions for the global cardinality constraint in the context of counting-based search. It first highlights an error and then presents a way to correct the upper bound on the number of solutions for this constraint

Exact methods for Bayesian network structure learning and cost function networks

Les modèles graphiques discrets représentent des fonctions jointes sur de grands ensembles de variables en tant qu'une combinaison de fonctions plus petites. Il existe plusieurs instanciations de modèles graphiques, notamment des modèles probabilistes et dirigés comme les réseaux Bayésiens, ou des modèles déterministes et non-dirigés comme les réseaux de fonctions de coûts. Des requêtes comme trouver l'explication la plus probable (MPE) sur un réseau Bayésiens, et son équivalent, trouver une solution de coût minimum sur un réseau de fonctions de coût, sont toutes les deux des tâches d’optimisation combinatoire NP-difficiles. Il existe cependant des techniques de résolution robustes qui ont une large gamme de domaines d'applications, notamment les réseaux de régulation de gènes, l'analyse de risques et le traitement des images. Dans ce travail, nous contribuons à l'état de l'art de l'apprentissage de la structure des réseaux Bayésiens (BNSL), et répondons à des requêtes de MPE et de minimisation des coûts sur les réseaux Bayésiens et les réseaux de fonctions de coûts. Pour le BNSL, nous découvrons un nouveau point dans l'espace de conception des algorithmes de recherche qui atteint un compromis différent entre la qualité et la vitesse de l'inférence. Les algorithmes existants optent soit pour la qualité maximale de l'inférence en utilisant la programmation linéaire en nombres entiers (PLNE) et la séparation et évaluation, soit pour la vitesse de l'inférence en utilisant la programmation par contraintes (PPC). Nous définissons des propriétés d'une classe spéciale d'inégalités, qui sont appelées "les inégalités de cluster" et qui mènent à un algorithme avec une qualité d'inférence beaucoup plus puissante que celle basée sur la PPC, et beaucoup plus rapide que celle basée sur la PLNE. Nous combinons cet algorithme avec des idées originales pour une propagation renforcée ainsi qu'une représentation de domaines plus compacte, afin d'obtenir des performances dépassant l'état de l'art dans le solveur open source ELSA (Exact Learning of bayesian network Structure using Acyclicity reasoning). Pour les réseaux de fonctions de coûts, nous identifions une faiblesse dans l'utilisation de la relaxation continue dans une classe spécifique de solveurs, y compris le solveur primé "ToulBar2". Nous prouvons que cette faiblesse peut entraîner des décisions de branchement sous-optimales et montrons comment détecter un ensemble maximal de telles décisions qui peuvent ensuite être évitées par le solveur. Cela permet à ToulBar2 de résoudre des problèmes qui étaient auparavant solvables uniquement par des algorithmes hybrides.Discrete Graphical Models (GMs) represent joint functions over large sets of discrete variables as a combination of smaller functions. There exist several instantiations of GMs, including directed probabilistic GMs like Bayesian Networks (BNs) and undirected deterministic models like Cost Function Networks (CFNs). Queries like Most Probable Explanation (MPE) on BNs and its equivalent on CFNs, which is cost minimisation, are NP-hard, but there exist robust solving techniques which have found a wide range of applications in fields such as bioinformatics, image processing, and risk analysis. In this thesis, we make contributions to the state of the art in learning the structure of BNs, namely the Bayesian Network Structure Learning problem (BNSL), and answering MPE and minimisation queries on BNs and CFNs. For BNSL, we discover a new point in the design space of search algorithms, which achieves a different trade-off between inference strength and speed of inference. Existing algorithms for it opt for either maximal strength of inference, like the algorithms based on Integer Programming (IP) and branch-and-cut, or maximal speed of inference, like the algorithms based on Constraint Programming (CP). We specify properties of a specific class of inequalities, called cluster inequalities, which lead to an algorithm that performs much stronger inference than that based on CP, much faster than that based on IP. We combine this with novel ideas for stronger propagation and more compact domain representations to achieve state-of-the-art performance in the open-source solver ELSA (Exact Learning of bayesian network Structure using Acyclicity reasoning). For CFNs, we identify a weakness in the use of linear programming relaxations by a specific class of solvers, which includes the award-winning open-source ToulBar2 solver. We prove that this weakness can lead to suboptimal branching decisions and show how to detect maximal sets of such decisions, which can then be avoided by the solver. This allows ToulBar2 to tackle problems previously solvable only by hybrid algorithms

Soft global constraints in constraint optimization and weighted constraint satisfaction.

Leung, Ka Lun.Thesis (M.Phil.)--Chinese University of Hong Kong, 2009.Includes bibliographical references (leaves 118-126).Abstract also in Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Constraint Satisfaction and Global Constraints --- p.3Chapter 1.2 --- Soft Constraints --- p.4Chapter 1.3 --- Motivation and Goal --- p.5Chapter 1.4 --- Outline of the Thesis --- p.6Chapter 2 --- Background --- p.8Chapter 2.1 --- Constraint Satisfaction Problems --- p.8Chapter 2.1.1 --- Backtracking Tree Search --- p.10Chapter 2.1.2 --- Local Consistency in CSP --- p.11Chapter 2.1.3 --- Constraint Optimization Problem --- p.16Chapter 2.2 --- Weighted Constraint Satisfaction --- p.21Chapter 2.2.1 --- Branch and Bound Search --- p.23Chapter 2.2.2 --- Local Consistency in WCSP --- p.26Chapter 2.3 --- Global Constraints --- p.35Chapter 2.4 --- Flow Theory --- p.37Chapter 3 --- Related Work --- p.39Chapter 3.1 --- Handling Soft Constraints in COPs --- p.39Chapter 3.2 --- Global Constraints --- p.40Chapter 3.2.1 --- Hard Global Constraints --- p.40Chapter 3.2.2 --- Soft Global Constraints --- p.41Chapter 3.3 --- Local Consistency in Weighted CSP --- p.42Chapter 4 --- “Soft as Hard´ح Approach --- p.44Chapter 4.1 --- The General “Soft as Hard´ح Approach --- p.44Chapter 4.2 --- Cost-based GAC --- p.49Chapter 4.3 --- Empirical Results --- p.53Chapter 5 --- Weighted CSP Approach --- p.55Chapter 5.1 --- Strong 0-Inverse Consistency --- p.55Chapter 5.1.1 --- 0-Inverse Consistency and Strong 0-Inverse Consistency --- p.56Chapter 5.1.2 --- Comparison with Other Consistencies --- p.62Chapter 5.2 --- Generalized Arc Consistency Star --- p.65Chapter 5.3 --- Full Directional Generalized Arc Consistency Star --- p.72Chapter 5.4 --- Generalizing EDAC* --- p.78Chapter 5.5 --- Implementation Issues --- p.87Chapter 6 --- Towards A Library of Efficient Soft Global Constraints --- p.90Chapter 6.1 --- The allDifferent Constraint --- p.91Chapter 6.1.1 --- All Interval Series --- p.93Chapter 6.1.2 --- Latin Square --- p.95Chapter 6.2 --- The GCC Constraint --- p.97Chapter 6.2.1 --- Latin Square --- p.100Chapter 6.2.2 --- Round Robin Tournament --- p.100Chapter 6.3 --- The Same Constraint --- p.102Chapter 6.3.1 --- Fair Scheduling --- p.104Chapter 6.3.2 --- People-Mission Scheduling --- p.105Chapter 6.4 --- The Regular Constraint --- p.106Chapter 6.4.1 --- Nurse Rostering Problem --- p.110Chapter 6.4.2 --- Modelling Stretch() Constraint --- p.111Chapter 6.5 --- Discussion --- p.113Chapter 7 --- Conclusion and Remarks --- p.115Chapter 7.1 --- Contributions --- p.115Chapter 7.2 --- Future Work --- p.117Bibliography --- p.11

Contrôle de la propagation et de la recherche dans un solveur de contraintes

La programmation par contraintes est souvent décrite, utopiquement, comme un paradigme déclaratif dans lequel l utilisateur décrit son problème et le solveur le résout. Bien entendu, la réalité des solveurs de contraintes est plus complexe, et les besoins de personnalisation des techniques de modélisation et de résolution évoluent avec le degré d expertise des utilisateurs. Cette thèse porte sur l enrichissement de l arsenal des techniques disponibles dans les solveurs de contraintes. D une part, nous étudions la contribution d un système d explications à l exploration de l espace de recherche, dans le cadre spécifique d une recherche locale. Deux heuristiques de voisinages génériques exploitant singulièrement les explications sont décrites. La première se base sur la difficulté de réparer une solution partiellement détruite, la seconde repose sur la nature non-optimale de la solution courante. Ces heuristiques mettent à jour la structure interne des problèmes traités pour construire des voisins de bonne qualité pour une recherche à voisinage large. Elles sont complémentaires d autres heuristiques de voisinages génériques, avec lesquels elles peuvent être combinées efficacement. De plus, nous proposons de rendre le système d explications paresseux afin d en minimiser l empreinte. D autre part, nous effectuons un état des lieux des savoir-faire relatifs aux moteurs de propagation pour les solveurs de contraintes. Ces données sont exploitées opérationnellement à travers un langage dédié qui permet de personnaliser la propagation au sein d un solveur, en fournissant des structures d implémentation et en définissant des points de contrôle dans le solveur. Ce langage offre des concepts de haut niveau permettant à l utilisateur d ignorer les détails de mise en œuvre du solveur, tout en conservant un bon niveau de flexibilité et certaines garanties. Il permet l expression de schémas de propagation spécifiques à la structure interne de chaque problème. La mise en œuvre et les expérimentations ont été effectués dans le solveur de contraintes Choco. Cette thèse a donné lieu à une nouvelle version de l outil globalement plus efficace et nativement expliqué.Constraint programming is often described, idealistically, as a declarative paradigm in which the user describes the problem and the solver solves it. Obviously, the reality of constraint solvers is more complex, and the needs in customization of modeling and solving techniques change with the level of expertise of users. This thesis focuses on enriching the arsenal of available techniques in constraint solvers. On the one hand, we study the contribution of an explanation system to the exploration of the search space in the specific context of a local search. Two generic neighborhood heuristics which exploit explanations singularly are described. The first one is based on the difficulty of repairing a partially destroyed solution, the second one is based on the non-optimal nature of the current solution. These heuristics discover the internal structure of the problems to build good neighbors for large neighborhood search. They are complementary to other generic neighborhood heuristics, with which they can be combined effectively. In addition, we propose to make the explanation system lazy in order to minimize its footprint. On the other hand, we undertake an inventory of know-how relative to propagation engines of constraint solvers. These data are used operationally through a domain specific language that allows users to customize the propagation schema, providing implementation structures and defining check points within the solver. This language offershigh-level concepts that allow the user to ignore the implementation details, while maintaining a good level of flexibility and some guarantees. It allows the expression of propagation schemas specific to the internal structure of each problem solved. Implementation and experiments were carried out in the Choco constraint solver, developed in this thesis. This has resulted in a new version of the overall effectiveness and natively explained tool.NANTES-ENS Mines (441092314) / SudocSudocFranceF

LOGIC AND CONSTRAINT PROGRAMMING FOR COMPUTATIONAL SUSTAINABILITY

Computational Sustainability is an interdisciplinary field that aims to develop computational and mathematical models and methods for decision making concerning the management and allocation of resources in order to help solve environmental problems. This thesis deals with a broad spectrum of such problems (energy efficiency, water management, limiting greenhouse gas emissions and fuel consumption) giving a contribution towards their solution by means of Logic Programming (LP) and Constraint Programming (CP), declarative paradigms from Artificial Intelligence of proven solidity. The problems described in this thesis were proposed by experts of the respective domains and tested on the real data instances they provided. The results are encouraging and show the aptness of the chosen methodologies and approaches. The overall aim of this work is twofold: both to address real world problems in order to achieve practical results and to get, from the application of LP and CP technologies to complex scenarios, feedback and directions useful for their improvement

Knowledge Components and Methods for Policy Propagation in Data Flows

Data-oriented systems and applications are at the centre of current developments of the World Wide Web (WWW). On the Web of Data (WoD), information sources can be accessed and processed for many purposes. Users need to be aware of any licences or terms of use, which are associated with the data sources they want to use. Conversely, publishers need support in assigning the appropriate policies alongside the data they distribute. In this work, we tackle the problem of policy propagation in data flows - an expression that refers to the way data is consumed, manipulated and produced within processes. We pose the question of what kind of components are required, and how they can be acquired, managed, and deployed, to support users on deciding what policies propagate to the output of a data-intensive system from the ones associated with its input. We observe three scenarios: applications of the Semantic Web, workflow reuse in Open Science, and the exploitation of urban data in City Data Hubs. Starting from the analysis of Semantic Web applications, we propose a data-centric approach to semantically describe processes as data flows: the Datanode ontology, which comprises a hierarchy of the possible relations between data objects. By means of Policy Propagation Rules, it is possible to link data flow steps and policies derivable from semantic descriptions of data licences. We show how these components can be designed, how they can be effectively managed, and how to reason efficiently with them. In a second phase, the developed components are verified using a Smart City Data Hub as a case study, where we developed an end-to-end solution for policy propagation. Finally, we evaluate our approach and report on a user study aimed at assessing both the quality and the value of the proposed solution

Consistency techniques for linear global cost functions in weighted constraint satisfaction.

在加權約束滿足問題中使用多元價值函數需要強大的一致相容性技術，而在多元價值函數中維護一致相容性並不是一項簡單的工作。能在多項式時間內找出多元價值函數的最少價值，而且不被投影及擴展操作所破壞，是讓該多元價值函數實用的主要條件。但是，有很多有用的多元價值函數尚未有多項式時間的算法找出其最少價值，因而未能在加權約束滿足問題中實用地使用它們。我們定義了一類可被建構為整數線性規劃的多元價值函數，並稱它們為多項式線性投影安全(PLPS)價值函數。該類價值函數的最少價值能由解答整數線性規劃中找出，而這個特性並不會被投影及擴展操作所影響。線性鬆馳能讓我們找出一個最少價值的接近值，並避免了解答整數線性規劃的NP-難困難性。該最少價值的接近值能作為最少價值的下限以供維護鬆馳一致相容性概念。在實踐中我們示範了使用PLPS價值函數的組合的好處。我們定義了整數多項式線性投影安全(IPLPS)價值函數作為PLPS價值函數的一個子類，並讓我們表示組合該類價值函數的好處。在一個加權約束滿足問題的一致相容性α中，我們表示了在IPLPS價值函數的組合中維護鬆馳α比在單獨的IPLPS價值函數中維護α強大。這結果可用在能在多項式時間中找出最少價值，但不能在多項式時間中找出它們的組合的最少價值的IPLPS價值函數中。基於流量投影安全(flow-based projection-safe)及可多項式分解(polynomially decomposable)價值函數的一個重要的子類屬於這一類的IPLPS價值函數。在實驗中我們展示了我們的方法的可行性和效率。無論在時間或搜索空間的改進上，與現有的方法相比，在使用PLPS價值函數的組合和 IPLPS價值函數的組合時我們觀察到一個數量級的改進。The solving of Weighted CSP (WCSP) with global cost functions relies on powerful consistency techniques, but enforcing these consistencies on global cost functions is not a trivial task. Lee and Leung suggest that a global cost function can be used practically if we can find its minimum cost and perform projections/extensions on it in polynomial time, and at the same time projections and extensions should not destroy those conditions. However, there are many useful cost functions with no known polynomial time algorithms to compute the minimum costs yet.We propose a special class of global cost functions which can be modeled as integer linear programs, called polynomially linear projection-safe (PLPS) cost functions. We show that their minimum cost can be computed by integer programming and this property is unaffected by projections/extensions. By linear relaxation we can avoid the possible NP-hard time taken to solve the integer programs, as the approximation of their actual minimum costs can be obtained to serve as a good lower bound in enforcing the relaxed forms of common consistencies.We show the benets of using the conjunctions of PLPS cost functions empir-ically in terms of runtime. We introduce integral polynomially linear projection-safe (IPLPS) cost functions as a subclass of PLPS cost functions whose allow us to characterize the benets of using the conjunctions of them. Given a standard WCSP consistency α, we give theorems showing that maintaining relaxed α on a conjunction of IPLPS cost functions is stronger than maintaining α on the individual cost functions. A useful application of our method is on some IPLPS global cost functions, whose minimum cost computations are tractable and yet those for their conjunctions are not. We show that an important subclass of flow-based projection-safe and polynomially decomposable cost functions falls into this category.Experiments are conducted to demonstrate the feasibility and efciency of our framework. We observe orders of magnitude in runtime and search space improvements by using the conjunctions of PLPS and IPLPS cost functions with relaxed consistencies when compared with the existing approaches.Detailed summary in vernacular field only.Detailed summary in vernacular field only.Detailed summary in vernacular field only.Detailed summary in vernacular field only.Shum, Yu Wai.Thesis (M.Phil.)--Chinese University of Hong Kong, 2012.Includes bibliographical references (leaves 87-92).Abstracts also in Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Weighted Constraint Satisfaction Problems --- p.2Chapter 1.2 --- Motivation and Goal --- p.2Chapter 1.3 --- Outline of the Thesis --- p.4Chapter 2 --- Related Work --- p.6Chapter 2.1 --- Soft Constraint Frameworks --- p.6Chapter 2.2 --- Integer Linear Programming --- p.8Chapter 2.3 --- Global Cost Functions in WCSP --- p.8Chapter 3 --- Background --- p.11Chapter 3.1 --- Weighted Constraint Satisfaction Problems --- p.11Chapter 3.1.1 --- Branch and Bound Search --- p.14Chapter 3.1.2 --- Local consistencies in WCSP --- p.15Chapter 3.1.3 --- Global Cost Functions --- p.30Chapter 3.2 --- Integer Linear Programming --- p.31Chapter 4 --- Polynomially Linear Projection-Safe Cost Functions --- p.33Chapter 4.1 --- Non-tractable Global Cost Functions in WCSPs --- p.34Chapter 4.2 --- Polynomially Linear Projection-Safe Cost Functions --- p.37Chapter 4.3 --- Relaxed Consistencies on Polynomially Linear Projection-Safe Cost Functions --- p.44Chapter 4.4 --- Conjoining Polynomially Linear Projection-Safe Cost Functions --- p.50Chapter 4.5 --- Modeling Global Cost Functions as Polynomially Linear Projection- Safe Cost Functions --- p.53Chapter 4.5.1 --- The SOFT SLIDINGSUM{U+1D48}{U+1D52}{U+1D9C} Cost Function --- p.53Chapter 4.5.2 --- The SOFT EGCC{U+1D5B}{U+1D43}{U+02B3} Cost Function --- p.54Chapter 4.5.3 --- The SOFT DISJUNCTIVE/CUMULATIVE Cost Function --- p.56Chapter 4.6 --- Implementation Issues --- p.59Chapter 4.7 --- Experimental Results --- p.60Chapter 4.7.1 --- Generalized Car Sequencing Problem --- p.62Chapter 4.7.2 --- Magic Series Problem --- p.63Chapter 4.7.3 --- Weighted Tardiness Scheduling Problem --- p.65Chapter 5 --- Integral Polynomially Linear Projection-Safe Cost Functions --- p.68Chapter 5.1 --- Integral Polynomially Linear Projection-Safe Cost Functions --- p.69Chapter 5.2 --- Conjoining Global Cost Functions as IPLPS --- p.72Chapter 5.3 --- Experimental Results --- p.76Chapter 5.3.1 --- Car Sequencing Problem --- p.77Chapter 5.3.2 --- Examination Timetabling Problem --- p.78Chapter 5.3.3 --- Fair Scheduling --- p.79Chapter 5.3.4 --- Comparing WCSP Approach with Integer Linear programming Approach --- p.81Chapter 6 --- Conclusions --- p.83Chapter 6.1 --- Contributions --- p.83Chapter 6.2 --- Future Work --- p.85Bibliography --- p.8