Nicolas Beldiceanu

IMT Atlantique | IMT · DAPI

Given a table with a minimal set of input columns that functionally determines an output column, we introduce a method that tries to gradually decompose the corresponding minimal functional dependency (mfd) to acquire a formula expressing the output column in terms of the input columns. A first key element of the method is to create sub-problems th...

Motivated by identifying equations to automate the discovery of conjectures about sharp bounds on combinatorial objects, we introduce a CP model to acquire Boolean-arithmetic equations (BAE) from a table providing sharp bounds for various combinations of parameters.Boolean-arithmetic expressions consist of simple arithmetic conditions (SAC) connect...

To automate the discovery of conjectures on combinatorial objects, we introduce the concept of a map of sharp bounds on characteristics of combinatorial objects, that provides a set of interrelated sharp bounds for these combinatorial objects. We then describe a Bound Seeker, a CP-based system, that gradually acquires maps of conjectures. The syste...

In this article, I show how Jacques Pitrat’s teachings and work have influenced my career and have fed my thoughts and questions in the field of constraint programming. After a brief reminder of the key ideas I adopted, I show how I used these ideas throughout my career and indicate why these ideas are still relevant today.

The European project ASSISTANT will provide a set of AI-based digital twins that helps process engineers and production planners to operate collaborative mixed-model assembly lines based on the data collected from IoT devices and external data sources. Such a tool will help planners to design the assembly line, plan the production, operate the line...

Many constraints restricting the result of some computations over an integer sequence can be compactly represented by counter automata. We improve the propagation of the conjunction of such constraints on the same sequence by synthesising a database of linear and non-linear invariants using their counter-automaton representation. The obtained invar...

For two families of time-series constraints with the aggregator and features and , we provide parameterised sharp lower and upper bounds on the sum of the time-series variables wrt these families of constraints. This is important in many applications, as this sum represents the cost, for example the energy used, or the manpower effort expended. We...

Given, a sequence $\mathcal{X}$ of $n$ variables, a time-series constraint ctr using the Sum aggregator, and a sliding time-series constraint enforcing the constraint ctr on each sliding window of $\mathcal{X}$ of $m$ consecutive variables, we describe a $\Theta(n)$ time complexity checker, as well as a $\Theta(n)$ space complexity reformulation fo...

Many constraints restricting the result of some computations over an integer sequence can be compactly represented by register automata. We improve the propagation of the conjunction of such constraints on the same sequence by synthesising a database of linear and non-linear invariants using their register-automaton representation. The obtained inv...

We introduce the concept of regular expression characteristics as a unified way to concisely express bounds on time-series constraints. This allows us not only to define time-series constraints in a compositional way, but also to deal with their combinatorial aspect in a compositional way, without developing ad-hoc bounds for each time-series const...

We propose a systematic approach for generating linear implied constraints that link the values returned by several automata with accumulators after consuming the same input sequence. The method handles automata whose accumulators are increased by (or reset to) some non-negative integer value on each transition. We evaluate the impact of the genera...

We consider, for an integer time series, two families of constraints restricting the max, and the sum, respectively, of the surfaces of the elements of the sub-series corresponding to occurrences of some pattern. In recent work these families were identified as the most difficult to solve compared to all other time-series constraints. For all patte...

For all 8192 combinations of Allen’s 13 relations between one task with origin \(o_i\) and fixed length \(\ell _i\) and another task with origin \(o_j\) and fixed length \(\ell _j\), this paper shows how to systematically derive a formula \(F(\underline{o_j}, \overline{o_j}, \ell _i, \ell _j)\), where \(\underline{o_j}\) and \(\overline{o_j}\) resp...

With the emergence of the Future Internet and the dawning of new IT models such as cloud computing, the usage of data centers (DC), and consequently their power consumption, increase dramatically. Besides the ecological impact, the energy consumption is a predominant criterion for DC providers since it determines the daily cost of their infrastruct...

We describe a system which generates finite domain constraint models from positive example solutions, for highly structured problems. The system is based on the global constraint catalog, providing the library of constraints that can be used in modeling, and the Constraint Seeker tool, which finds a ranked list of matching constraints given one or...

Integer time series are often subject to constraints on the aggregation of the integer features of all occurrences of some pattern within the series. For example, the number of inflexions may be constrained, or the sum of the peak maxima, or the minimum of the peak widths. It is currently unknown how to maintain domain consistency efficiently on su...

We introduce a propagator for pairs of Sum constraints, where the expressions in the sums respect a form of convexity. This propagator is parametric and can be instantiated for various concrete pairs, including Deviation, Spread, and the conjunction of Linear≤ and Among. We show that despite its generality, our propagator is competitive in theory a...

A checker for a constraint on a variable sequence can often be compactly specified by an automaton, possibly with accumulators, that consumes the sequence of values taken by the variables; such an automaton can also be used to decompose its specified constraint into a conjunction of logical constraints. The inference achieved by this decomposition...

Given a sequence of tasks \(\mathcal {T}\) subject to precedence constraints between adjacent tasks, and given a set of fixed intervals \(\mathcal {I}\), the TaskIntersection \((\mathcal {T},\mathcal {I},o, inter )\) constraint restricts the overall intersection of the tasks of \(\mathcal {T}\) with the fixed intervals of \(\mathcal {I}\) to be gre...

For all 8192 combinations of Allen's 13 relations between one task with origin o i and fixed length i and another task with origin o j and fixed length j , we give a formula F(o j , o j , i , j), where o j and o j respectively denote the earliest and the latest origin of task j, evaluating to a set of integers which are infeasible for o i for the g...

We describe a large family of constraints for structural time series by means of function composition. These constraints are on aggregations of features of patterns that occur in a time series, such as the number of its peaks, or the range of its steepest ascent. The patterns and features are usually linked to physical properties of the time series...

With the emergence of the Future Internet and the dawning of new IT models such as cloud computing, the usage of data centers (DC), and consequently their power consumption, increase dramatically. Besides the ecological impact, the energy consumption is a predominant criteria for DC providers since it determines the daily cost of their infrastructu...

This paper introduces a family of synchronized sweep-based filtering algorithms for handling scheduling problems involving resource and precedence constraints. The key idea is to filter all constraints of a scheduling problem in a synchronized way in order to scale better. In addition to normal filtering mode, the algorithms can run in greedy mode,...

Consider a constraint on a sequence of variables functionally determining a result variable that is unchanged under reversal of the sequence. Most such constraints have a compact encoding via an automaton augmented with accumulators, but it is unknown how to maintain domain consistency efficiently for most of them. Using such an automaton for such...

Euler, Vandermonde, Dudeney, Schwesk, Berliner, Conrad and many others already considered knight's tours on chessboards. The classical knight's tour problem consist of finding out on a N × M chessboard a sequence of legal knight moves that visit every cell exactly once and finish by returning to the initial cell. A more challenging question is to g...

We present a few challenges that we consider important to tackle for the future of constraint programming. The focus is put on simplifying the design and implementation of propagators in solvers.

Constraints over finite sequences of variables are ubiquitous in sequencing and timetabling. Moreover, the wide variety of such constraints in practical applications led to general modelling techniques and generic propagation algorithms, often based on deterministic finite automata (DFA) and their extensions. We consider counter-DFAs (cDFA), which...

We present an application to extract and solve constraint models from sample solutions of the Unit Commitment Problem of EDF, which computes the power output for each power plant in France as a 48 hour time series. Our aim is to describe and automatically generate the plant-specific model constraints common to the optimal solutions obtained over mu...

We introduce a propagator for abstract pairs of Sum constraints, where the expressions in the sums respect a form of convexity. This propagator is parametric and can be instantiated for various concrete pairs, including Deviation, Spread, and the conjunction of Sum and Count. We show that despite its generality, our propagator is competitive in the...

This paper presents a sweep based algorithm for the k-dimensional cumulative constraint, which can operate in filtering mode as well as in greedy assignment mode. Given n tasks and k resources, this algorithm has a worst-case time complexity of O(kn 2) but scales well in practice. In greedy assignment mode, it handles up to 1 million tasks with 64...

We provide a filtering algorithm achieving GAC for the conjunction of constraints atleast(b, (x0, x1,., xn-1), V) and Σi=0n-1 ai xi ≤ c, where the at least constraint enforces b variables out of x0, x 1,., xn-1 to be assigned to a value in the set V. This work was motivated by learning simple polynomials, i.e. finding the coefficients of polynomial...

We introduce a simple idea for deriving reified global constraints in a systematic way. It is based on the observation that most global constraints can be reformulated as a conjunction of total function constraints together with a constraint that can be easily reified.

Matrix models are ubiquitous for constraint problems. Many such problems have a matrix of variables $\mathcal{M}$ , with the same constraint C defined by a finite-state automaton $\mathcal{A}$ on each row of $\mathcal{M}$ and a global cardinality constraint $\mathit{gcc}$ on each column of $\mathcal{M}$ . We give two methods for deriving,...

We describe a system which generates finite domain constraint models from positive example solutions, for highly structured problems. The system is based on the global constraint catalog, providing the library of constraints that can be used in modeling, and the Constraint Seeker tool, which finds a ranked list of matching constraints given one or...

This paper gives an O(n log n) bound-consistency filtering algorithm for the conjunction alldifferent(V0, V1,⋯, V n-1) A f(V0) ⊕ f(V1) ⊕ ··· ⊕ f(Vn-1) ≤ cst, (V0, V1,⋯, Vn-1, cst ∈ N+), where (ℕ, ⊕) is a commutative group, f is a unary function, and both ⊕; and f are monotone increasing. This complexity is equal to the complexity of the bound-consi...

We present a system which generates global constraint models from few positive examples of problem solutions. In contrast to previous constraint acquisition work, we present a novel approach based on the global constraint catalog and the Constraint Seeker tool which generates models for problems which can be expressed as regular conjunctions of sim...

This paper presents a sweep based algorithm for the cumulative constraint, which can operate in filtering mode as well as in greedy assignment mode. Given n tasks, this algorithm has a worst-case time complexity of O(n 2). In practice, we use a variant with better average-case complexity but worst-case complexity of O(n 2 logn), which goes down to...

The 9th International Conference on Integration of Artificial Intelligence and Operations Research Techniques in Constraint Programming was held in Nantes, France, May 28-June 1, 2012. The aim of the CPAIOR conference series is to bring together interested researchers from constraint programming (CP), artificial intelligence (AI), and operations re...

This paper introduces the SEQ BIN meta-constraint with a polytime algorithm achieving general- ized arc-consistency according to some properties. SEQ BIN can be used for encoding counting con- straints such as CHANGE, SMOOTH or INCREAS- ING NVALUE. For some of these constraints and some of their variants GAC can be enforced with a time and space co...

Given a sequence of variables X = 〈x 0, x 1, …, x n − 1 〉, we consider the IncreasingSum constraint, which imposes ∀ i ∈ [0, n − 2] x i ≤ x i + 1, and \(\sum_{x_i \in X} x_i = s\). We propose an Θ(n) bound-consistency algorithm for IncreasingSum.

Many cumulative problems are such that the horizon is fixed and cannot be delayed. In this situation, it often occurs that all the activities cannot be scheduled without exceeding the capacity at some points in time. Moreover, this capacity is not necessarily always the same during the scheduling period. This article introduces a new constraint for...

In this paper we describe a Constraint Seeker application which provides a web interface to search for global constraints in the global constraint catalog, given positive and negative, fully instantiated (ground) examples. Based on the given instances the tool returns a ranked list of matching constraints, the rank indicating whether the constraint...

Considering problems that have a strong internal structure, this pa-per shows how to generate constraint models from a set of positive, flat samples (i.e., solutions) without knowing a priori neither the constraint candidates, nor the way variables are shared within constraints. We describe two key contributions to building such a model generator:...

This paper introduces the SEQ BIN meta-constraint with a polytime algorithm achieving generalized arc-consistency. SEQ BIN can be used for encoding counting constraints such as CHANGE, SMOOTH, or INCREASING NVALUE. For all of them the time and space complexity is linear in the sum of domain sizes, which improves or equals the best known results of...

This article describes new filtering methods for the cumulative constraint. The first method introduces the so called longest closed hole and longest open hole problems. For these two problems it first provides bounds and exact methods and then shows how to use them in the context of the non-overlapping constraint. The second method introduces bala...

In this paper we give an overview of an early prototype which learns structured constraint models from flat, positive examples of solutions. It is based on previous work on a Constraint Seeker, which finds constraints in the global constraint catalog satisfying positive and negative examples. In the current tool we extend this system to find struct...

In this note we describe the solution of perfect square placement problems with CHIP. A set of 207 perfect square placement problems from [BD92] is used. We present the constraint model in CHIP, which uses the global constraints diffn and cumulative together with a specific labeling routine for perfect placement problems. We show that all problems...

The geost constraint has been proposed to model and solve discrete placement problems involving multi-dimensional boxes (packing in space and time). The filtering technique is based on a sweeping algorithm that requires the ability for each constraint to compute a forbidden box around a given fixed point and within a surrounding area. Several cases...

Matrix models are ubiquitous for constraint problems. Many such problems have a matrix of variables M\mathcal{M}, with the same constraint defined by a finite-state automaton A\mathcal{A} on each row of M\mathcal{M} and a global cardinality constraint gcc{\mathit{gcc}} on each column of M\mathcal{M}. We give two methods for deriving, by double coun...

This paper introduces the Increasing_Nvalue constraint, which restricts the number of distinct values assigned to a sequence of variables so that each variable in the sequence is less than or equal to its successor. This constraint is a specialization of the Nvalue constraint, motivated by symmetry breaking. Propagating the Nvalue constraint is kno...

Cet article introduit la contrainte Increasing NValue, qui restreint le nombre de valeurs distinctes affectées à une séquence de variables, de sorte que chaque variable de la séquence soit inférieure ou égale à la variable la succédant immédiatement. Cette contrainte est une spécialisation de la contrainte NValue, motivée par le besoin de casser de...

This paper presents the constraint class seq bin(N;X;C;B) where N is an integer variable, X is a sequence of integer variables and C and B are two binary constraints. A constraint of the seq bin class enforces the two following conditions: (1) N is equal to the number of times that the constraint C is satised on two consecutive variables in X, and...

http://www.emn.fr/x-info/sdemasse/gccat/

We present two constraints that partition the vertices of an undirected n-vertex, m-edge graph G=( V, E)\mathcal {G}=( \mathcal {V}, \mathcal {E}) into a set of vertex-disjoint trees. The first is the resource-forest constraint, where we assume that a subset R Í V\mathcal {R}\subseteq \mathcal {V} of the vertices are resource vertices. The cons...

This paper introduces six ways for handling a chain of lexicographic ordering (lex-chain) constraint between the origins of identical orthotopes (e.g., rectangles, boxes, hyper-rectangles) subject to the fact that they should not pairwise overlap. While the first two ways deal with the integration of a lex-chain constraint within a generic geometri...

Six Ways of Integrating Symmetries within Non-Overlapping Constraints

Compiling Business Rules in a Geometric Constraint over k-Dimensional Objects and Shapes

The tree constraint partitions a directed graph into node-disjoint trees. In many practical applications that involve such a partition, there exist side constraints specifying requirements on tree count, node degrees, or precedences and incomparabilities within node subsets. We present a generalisation of the tree constraint that incorporates such...

This paper presents a global constraint that enforces rules written in a language based on arithmetic and first-order logic to hold among a set of objects. In a first step, the rules are rewritten to Quantifier-Free Presburger Arithmetic (QFPA) formulas. Secondly, such formulas are compiled to generators of k-dimensional forbidden sets. Such genera...

This paper describes new filtering methods for the cumulative constraint. The first method introduces bounds for the so called longest cumulative hole problem and shows how to use these bounds in the context of the non-overlapping constraint. The second method introduces balancing knapsack constraints which relate the total height of the tasks that...

Within the framework of continuous and multi- resources cumulative constraints, a task T expresses a piecewise linear resource function and is represented by a sequence of p contiguous trapezoid sub-tasks with variable durations and heights. In this context, this pa- per provides an algorithm in O(p) for filtering the re- source assignment and the...

Programmation par Contraintes

This paper introduces a geometrical constraint kernel for handling the location in space and time of polymorphic k-dimensional objects subject to various geometrical and time constraints. The constraint kernel is generic in the sense that one of its parameters is a set of constraints on subsets of the objects. These constraints are handled globally...

This article first introduces an extension of the classical cumulative constraint: each task is no more a rectangle but rather a sequence of contiguous trapezoid sub-tasks with variable duration and heights. The resource function is no more constant but is a positive or negative piecewise linear function of time. Finally, a task is no more pre-assi...

Etant donné un graphe orienté $\cG$, le problème des $\NPATH$-chemins disjoints consiste à trouver une partition de $\cG$ en $\NPATH$ chemins noeuds-disjoints, tel que chaque chemin termine sur un noeud appartenant à un sous-ensemble des noeuds de $\cG$. Cet article fournit une condition nécessaire, pour le problème des $\NPATH$-chemins disjoints,...

Cet article introduit une extension de la contrainte cumulative classique : une tâche n'est plus représentée par un simple rectangle mais par une suite de sous-tâches trapézoïdales de durées et hauteurs variables. La fonction de ressource n'est plus une constante, mais une fonction du temps, linéaire par morceaux, positive ou négative. Enfin, unetâ...

Given a directed graph G, the K node-disjoint paths problem consists in finding a partition of G into K node-disjoint paths, such that each path ends up in a given subset of nodes in G. This article provides a necessary condition for the K node-disjoint paths problem which combines (1) the structure of the reduced graph associated with G, (2) the s...

The catalogue of global constraints is reviewed, focusing on the graph-based de- scription of global constraints. A number of possible enhancements are proposed as well as several research paths for the development of the area.

We study the "inter-distance constraint," also known as the global minimum distance constraint, that ensures that the distance between any pair of variables is at least equal to a given value. When this value is 1, the inter-distance constraint reduces ...

This paper introduces a geometrical constraint kernel for handling the location in space and time of polymorphic k-dimensional objects subject to various geometrical and time constraints. The constraint kernel is generic in the sense that one of its parameters is a set of constraints on subsets of the objects. These constraints are handled globally...

This article presents a basic scheme for deriving systematically a filtering algorithm from the graph properties based representation of global constraints. This scheme is based on the bounds of the graph parameters used in the description of a global constraint. The article provides bounds for the most common used graph parameters.

This paper presents a new generic filtering algorithm which simultaneously considers nn conjunctions of constraints as well as those constraints mentioning some variables YkYk of the pairs X,Yk(1⩽k⩽n) occurring in these conjunctions. The main benefit of this new technique comes from the fact that, for adjusting the bounds of a variable X according...

This article presents a generic filtering scheme, based on th e graph de- scription of global constraints. This description is define d by a network of binary constraints and a list of elementary graph properties: each solution of the global constraint corresponds to a subgraph of the initial network, retaining only the sat- isfied binary constrain...

Nous présentons deux contraintes qui partitionnent les sommets d'un graphe non-orienté G = (V, E), où |V| = n et |E| = m, en un ensemble d'arbres disjoints. La première contrainte, resource-forest, spécifie que chaque arbre dans la forêt doit contenir au moins un sommet ressource. L'ensemble des ressources est un sous-ensemble R ⊆ V. Cette contrain...

Nous présentons une contrainte de partitionnement de graphe par des arbres sous contraintes de degré. Étant donné un graphe orienté G, la contrainte spécifie que toute solution est formée par N arbres, tels que le nombre de fils d'un sommet quelconque de G appartienne à un ensemble donné de valeurs. Il existe pour la contrainte d'arbre pure, introd...

We present two constraints that partition the vertices of an undirected n-vertex, m-edge graph G=(V,E){\mathcal{G}}=({\mathcal{V}},{\mathcal{E}}) into a set of vertex-disjoint trees. The first is the resource-forest constraint, where we assume that a subset R Í V{\mathtt{R}}\subseteq {\mathcal{V}} of the vertices are resource vertices. The constrai...

The tree and path constraints, for digraph partitioning by vertex-disjoint trees and paths respectively, are unified within a single global constraint, including a uniform treatment of a variety of useful side constraints, such as precedence, incomparability, and degree constraints. The approach provides a sharp improvement over an existing path co...

Cet article présente un schéma de filtrage générique, basé sur la description de contraintes globales sous la forme de propriétés de graphes. Cette description est définie par un réseau de contraintes binaires et une liste de propriétés de graphe élémentaires : chaque solution de la contrainte globale correspond à un sous-graphe du réseau initial,...

Graph-Properties Based Filtering

Global Constraints: Introduction and Graph-Based Representation.

Cet article présente un schéma de filtrage générique, basé sur la description de contraintes globales sous la forme de propriétés de graphes. Cette description est définie par un réseau de contraintes binaires et une liste de propriétés de graphe élémentaires : chaque solution de la contrainte globale correspond à un sous-graphe du réseau initial,...

We define the \Same\ and \UsedBy\ constraints. \UsedBy\ takes two sets of variables $X$ and $Z$ such that $|X|\ge |Z|$ and assigns values to them such that the multiset of values assigned to the variables in $Z$ is contained in the multiset of values assigned to the variables in $X$. \Same\ is the special case of \UsedBy\ in which $|X|=|Z|$. We sho...

This article deals with global constraints for which the set of solu- tions can be recognized by an extended nite automaton whose size is bounded by a polynomial in n, where n is the number of variables of the corresponding global constraint. By reducing the automaton to a conjunction of signature and transition constraints we show how to systemati...

This report presents a basic scheme,for deriving systematically a ltering algorithm from the graph properties based representation of global constraints. This scheme,is based on the bounds of the graph characteristics used in the description of a global constraint. The report provides bounds,for the most common,used graph characteristics.

This report presents a database of about graph invariants for de- riving systematically necessary conditions from the graph properties based rep- resentation of global constraints. This scheme is based on invariants on the graph characteristics used in the description of a global constraint. A SICStus Prolog implementation based on arithmetic and l...