A Comparison of Lex Bounds for Multiset Variables in Constraint Programming

Set and multiset variables in constraint programming have typically been represented using subset bounds. However, this is a weak representation that neglects potentially useful information about a set such as its cardinality. For set variables, the length-lex (LL) representation successfully provides information about the length (cardinality) and position in the lexicographic ordering. For multiset variables, where elements can be repeated, we consider richer representations that take into account additional information. We study eight different representations in which we maintain bounds according to one of the eight different orderings: length-(co)lex (LL/LC), variety-(co)lex (VL/VC), length-variety-(co)lex (LVL/LVC), and variety-length-(co)lex (VLL/VLC) orderings. These representations integrate together information about the cardinality, variety (number of distinct elements in the multiset), and position in some total ordering. Theoretical and empirical comparisons of expressiveness and compactness of the eight representations suggest that length-variety-(co)lex (LVL/LVC) and variety-length-(co)lex (VLL/VLC) usually give tighter bounds after constraint propagation. We implement the eight representations and evaluate them against the subset bounds representation with cardinality and variety reasoning. Results demonstrate that they offer significantly better pruning and runtime.Comment: 7 pages, Proceedings of the Twenty-Fifth AAAI Conference on Artificial Intelligence (AAAI-11

A study on set variable representations in constraint programming

Il lavoro presentato in questa tesi si colloca nel contesto della programmazione con vincoli, un paradigma per modellare e risolvere problemi di ricerca combinatoria che richiedono di trovare soluzioni in presenza di vincoli. Una vasta parte di questi problemi trova naturale formulazione attraverso il linguaggio delle variabili insiemistiche. Dal momento che il dominio di tali variabili può essere esponenziale nel numero di elementi, una rappresentazione esplicita è spesso non praticabile. Recenti studi si sono quindi focalizzati nel trovare modi efficienti per rappresentare tali variabili. Pertanto si è soliti rappresentare questi domini mediante l'uso di approssimazioni definite tramite intervalli (d'ora in poi rappresentazioni), specificati da un limite inferiore e un limite superiore secondo un'appropriata relazione d'ordine. La recente evoluzione della ricerca sulla programmazione con vincoli sugli insiemi ha chiaramente indicato che la combinazione di diverse rappresentazioni permette di raggiungere prestazioni di ordini di grandezza superiori rispetto alle tradizionali tecniche di codifica. Numerose proposte sono state fatte volgendosi in questa direzione. Questi lavori si differenziano su come è mantenuta la coerenza tra le diverse rappresentazioni e su come i vincoli vengono propagati al fine di ridurre lo spazio di ricerca. Sfortunatamente non esiste alcun strumento formale per paragonare queste combinazioni. Il principale obiettivo di questo lavoro è quello di fornire tale strumento, nel quale definiamo precisamente la nozione di combinazione di rappresentazioni facendo emergere gli aspetti comuni che hanno caratterizzato i lavori precedenti. In particolare identifichiamo due tipi possibili di combinazioni, una forte ed una debole, definendo le nozioni di coerenza agli estremi sui vincoli e sincronizzazione tra rappresentazioni. Il nostro studio propone alcune interessanti intuizioni sulle combinazioni esistenti, evidenziandone i limiti e svelando alcune sorprese. Inoltre forniamo un'analisi di complessità della sincronizzazione tra minlex, una rappresentazione in grado di propagare in maniera ottimale vincoli lessicografici, e le principali rappresentazioni esistenti

Modelling dynamic programming-based global constraints in constraint programming

Dynamic Programming (DP) can solve many complex problems in polynomial or pseudo-polynomial time, and it is widely used in Constraint Programming (CP) to implement powerful global constraints. Implementing such constraints is a nontrivial task beyond the capability of most CP users, who must rely on their CP solver to provide an appropriate global constraint library. This also limits the usefulness of generic CP languages, some or all of whose solvers might not provide the required constraints. A technique was recently introduced for directly modelling DP in CP, which provides a way around this problem. However, no comparison of the technique with other approaches was made, and it was missing a clear formalisation. In this paper we formalise the approach and compare it with existing techniques on MiniZinc benchmark problems, including the flow formulation of DP in Integer Programming. We further show how it can be improved by state reduction methods

Computing policy parameters for stochastic inventory control using stochastic dynamic programming approaches

The objective of this work is to introduce techniques for the computation of optimal and near-optimal inventory control policy parameters for the stochastic inventory control problem under Scarf’s setting. A common aspect of the solutions presented herein is the usage of stochastic dynamic programming approaches, a mathematical programming technique introduced by Bellman. Stochastic dynamic programming is hybridised with branch-and-bound, binary search, constraint programming and other computational techniques to develop innovative and competitive solutions. In this work, the classic single-item, single location-inventory control with penalty cost under the independent stochastic demand is extended to model a fixed review cost. This cost is charged when the inventory level is assessed at the beginning of a period. This operation is costly in practice and including it can lead to significant savings. This makes it possible to model an order cancellation penalty charge. The first contribution hereby presented is the first stochastic dynamic program- ming that captures Bookbinder and Tan’s static-dynamic uncertainty control policy with penalty cost. Numerous techniques are available in the literature to compute such parameters; however, they all make assumptions on the de- mand probability distribution. This technique has many similarities to Scarf’s stochastic dynamic programming formulation, and it does not require any ex- ternal solver to be deployed. Memoisation and binary search techniques are deployed to improve computational performances. Extensive computational studies show that this new model has a tighter optimality gap compared to the state of the art. The second contribution is to introduce the first procedure to compute cost- optimal parameters for the well-known (R, s, S) policy. Practitioners widely use such a policy; however, the determination of its parameters is considered com- putationally prohibitive. A technique that hybridises stochastic dynamic pro- gramming and branch-and-bound is presented, alongside with computational enhancements. Computing the optimal policy allows the determination of op- timality gaps for future heuristics. This approach can solve instances of consid- erable size, making it usable by practitioners. The computational study shows the reduction of the cost that such a system can provide. Thirdly, this work presents the first heuristics for determining the near-optimal parameters for the (R,s,S) policy. The first is an algorithm that formally models the (R,s,S) policy computation in the form of a functional equation. The second is a heuristic formed by a hybridisation of (R,S) and (s,S) policy parameters solvers. These heuristics can compute near-optimal parameters in a fraction of time compared to the exact methods. They can be used to speed up the optimal branch-and-bound technique. The last contribution is the introduction of a technique to encode dynamic programming in constraint programming. Constraint programming provides the user with an expressive modelling language and delegates the search for the solution to a specific solver. The possibility to seamlessly encode dynamic programming provides new modelling options, e.g. the computation of optimal (R,s,S) policy parameters. The performances in this specific application are not competitive with the other techniques proposed herein; however, this encoding opens up new connections between constraint programming and dynamic programming. The encoding allows deploying DP based constraints in modelling languages such as MiniZinc. The computational study shows how this technique can outperform a similar encoding for mixed-integer programming

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

Global Constraint Catalog, 2nd Edition (revision a)

This report presents a catalogue of global constraints where each constraint is explicitly described in terms of graph properties and/or automata and/or first order logical formulae with arithmetic. When available, it also presents some typical usage as well as some pointers to existing filtering algorithms