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Since their beginning in constraint programming, set solvers have been applied to a wide range of combinatorial search problems,
such as bin-packing, set partitioning, circuit and combinatorial design. In this paper we present and evaluate a new means
towards improving the practical reasoning power of Finite Set (FS) constraint solvers to better ad...
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The aggregation of generic items into coalitions leads to the creation of sets of homogenous entities. In this paper we accomplish this for an input set of arguments, and the result is a partition according to distinct lines of thought, i.e., groups of “coherent” ideas. We extend Dung's Argumentation Framework AF in order to deal with coalitions of...
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... More recently, other approaches were proposed to deal with finite sets of integers. Common ways to represent the domain of a set variable are the subset-bounds representation [17] and the lexicographic-bounds representation [163]. A radically different approach based on the Reduced Ordered Binary ...
Recent research has shown that the performance of a single, arbitrarily efficient algorithm can be significantly outperformed by using a portfolio of —possibly on-average slower— algorithms. Within the Constraint Programming (CP) context, a portfolio solver can be seen as a particular constraint solver that exploits the synergy between the constituent solvers of its portfolio for predicting which is (or which are) the best solver(s) to run for solving a new, unseen instance. In this thesis we examine the benefits of portfolio solvers in CP. Despite portfolio approaches have been extensively studied for Boolean Satisfiability (SAT) problems, in the more general CP field these techniques have been only marginally studied and used. We conducted this work through the investigation, the analysis and the construction of several portfolio approaches for solving both satisfaction and optimization problems. We focused in particular on sequential approaches, i.e., single-threaded portfolio solvers always running on the same core. We started from a first empirical evaluation on portfolio approaches for solving Constraint Satisfaction Problems (CSPs), and then we improved on it by introducing new data, solvers, features, algorithms, and tools. Afterwards, we addressed the more general Constraint Optimization Problems (COPs) by implementing and testing a number of models for dealing with COP portfolio solvers. Finally, we have come full circle by developing sunny-cp: a sequential CP portfolio solver that turned out to be competitive also in the MiniZinc Challenge, the reference competition for CP solvers.
... The task is to allocate clients on (possibly multiple) rings, satisfying the bandwidth constraints on the rings and minimizing the equipment cost. This problem has been tackled previously using mixed integer programming (MIP) [1] and constraint programming (CP) [2,3]. Much attention was devoted to variable branching heuristics and breaking ring symmetries (since all rings are identical). ...
... We also assume that the domains will remain the bound-consistent after any operation. 3 The filtering algorithm reasons on the common region in which possible sets of two domains overlaps. It tries to construct a feasible assignment and determines Algorithm 1 bc sbc nonEmptyIntersection (X sbc = sbc R X , P X , c X , c X , Y sbc ) ...
... Please refer to Section 5 in [2] for a detailed discussion. Another CP model was proposed in [3] and it broke symmetries by adding lexicographic constraints on set variables. With the additional lexicographic component, the solver obtained a tighter approximation of the set-variable domains. ...
This paper reconsiders the deployment of synchronous op-tical networks (SONET), an optimization problem naturally expressed in terms of set variables. Earlier approaches, using either MIP or CP technologies, focused on symmetry breaking, including the use of SBDS, and the design of effective branching strategies. This paper advocates an orthogonal approach and argues that the thrashing behavior experi-enced in earlier attempts is primarily due to a lack of pruning. It studies how to improve domain filtering by taking a more global view of the application and imposing redundant global constraints. The technical results include novel hardness results, propagation algorithms for global constraints, and inference rules. The paper also evaluates the contribu-tions experimentally by presenting a novel model with static symmetric-breaking constraints and a static variable ordering which is many orders of magnitude faster than existing approaches.
... Many of these new representations are based on combining together existing representations (e.g. [1,5,14,11]). For instance, the Cardinal set solver [1] reasons simultaneously about subset and cardinality bounds on the set. ...
... For example, we prove that the lengthlex representation for set variables (which combines lexicographic ordering and cardinality information and is designed to exploit cardinality and ordering constraints) is incomparable to a representation for set variables that deal with the lexicographic ordering and cardinality information separately or simultaneously. As a second example, we prove that the hybrid representation of [14] (which combines lexicographic, subset and cardinality bounds) is tighter than one that looks at pairs of bounds at the same time, but looser than one that looks at all the bounds simultaneously. ...
... That is, in the representation T , D T (S) = {s | lb T (S) ≤ T s ≤ T ub T (S)}. This is the case in [5,14] where the sets are ordered lexicographically, though in two different ways. We here call minlex (also denoted by M L) the total ordering in which sets are ordered by ≤ M L defined as follows: elements in the sets are ordered from smallest to largest and the lists obtained in this way are ordered lexicographically. ...
Within constraint programming, a number of different repre-sentations have been used for set variables. Many of these representations are based on combining together existing representations. To understand such combinations, we provide a formal definition of this notion of com-bination, and of the synchronisation between representations. We char-acterize two types of combinations, define bound consistency on these combinations and provide some tractability results. Finally, we compare the strength of the different combinations, positioning existing represen-tations within our taxonomy. This systematic approach opens the door to interesting new representations of set variables. Our results apply also to multiset variable representations.
... To obtain a set of benchmarks we increase n until the problem is only just satisfiable, so a benchmark is characterised by the 4 numbers n, l, d, w. There are 11 hard instances taken from [37]; 4 of these were too hard for our approach but we show results for the remaining 7 in Table 4. ...
The presence of symmetry in constraint satisfaction problems can cause a great deal of wasted search effort, and several methods for breaking symmetries have been reported. In this paper we describe a new method called Symmetry Breaking by Nonstationary Optimisation, which interleaves local search in the symmetry group with backtrack search on the constraint problem. It can be tuned to break each symmetry with an arbitrarily high probability with high runtime overhead, or as a lightweight but still powerful method with low runtime overhead. It has negligible memory requirement, it combines well with static lex-leader constraints, and its benefit increases with problem hardness.
... Cette représentation est simple et compacte mais a une capacité de représentation limitée. D'autres représentations approximatives [30, 61] ou exactes [67] ont été proposées. ...
Résumé L'interprétation de scènes complexes nécessite des connaissances sur les objets constituant la scène et leur organisation spatiale. Dans ce cadre, nous proposons une méthode de segmentation et de recon-naissance simultanées d'objets dans des images, s'appuyant sur une représentation structurelle de la scène et sur une méthode de propa-gation de contraintes. Les contributions de cet article sont illustrées sur l'exemple de la reconnaissance de structures du cerveau dans des images IRM. Le modèle structurel est un graphe décrivant ces struc-tures, leur apparence et leurs relations spatiales représentées par des modèles flous. La méthode de résolution proposée est une méthode glo-bale originale qui recherche une solution (c'est-à-dire l'affectation de régions de l'espace aux structures anatomiques du modèle) satisfaisant les relations du modèle structurel. Nous proposons de réduire progres-sivement l'espace de solutions par l'exclusion des affectations inco-hérentes avec un réseau de contraintes construit à partir du modèle structurel. La segmentation finale de chaque structure est réalisée par l'extraction d'une surface minimale. Mots-clés : Propagation de contraintes, relations spatiales floues, reconnaissance de structures dans les images, imagerie cérébrale.
... The HDR thesis of Gervet [57] contains a lot of interesting ideas and is certainly the best reference on this topic. Some information can be found in [125, 150, 58, 56, 126]. @BULLET Open Constraints. ...
Constraint programming (CP) is mainly based on filtering algorithms; their association with global constraints is one of the
main strengths of CP because they exploit the specific structure of each constraint. This chapter is an overview ofthese two
techniques. A collection of the most frequently used global constraints is given and some filtering algorithms are detailed.
In addition, we try to identify how filtering algorithms can be designed. At last, we identify some problems that deserve
to be addressed in the future.
... Padberg, 1976) – has been the transportation industry, dealing with problems of dispatching vehicles (Desaulniers et al., 2003; Westphal and Krumke, 2008), determining vehicle fleet mixes (Lee et al., 2008), routing and scheduling of vehicles (Alvarenga et al., 2007; Baldacci et al., 2008; Freling et al., 2003; Hong et al., 2009; Ileri et al., 2006; Jepsen et al., 2008; Kliewer et al., 2006) and scheduling of transportation system crews (Medard and Sawhney, 2007; Mesquita and Paias, 2008) among many others. Recent applications outside the transportation industry include problems of cellular manufacturing system design (Mahdavi et al., 2006), communication system network design (Oliveira et al., 2005 ('clique' form); Tombus and Bilgic, 2004), computer hardware and software designs (Bryson and Joseph, 2006; Thomadsen and Larsen, 2007), design of balanced student teams (Desrosiers et al., 2005), design of maintenance schedules for production machines (Grigoriev et al., 2006), facility location (Berger et al., 2007), image compression (Jyotheswar and Mahapatra, 2007), molten iron allocation in the steel industry (Tang et al., 2007), supply chain design (Chiang and Russell, 2004; Sadler and Gervet, 2008; Sindhuchao et al., 2005; Teo and Shu, 2004), 'surgical theater' planning involving room and surgical team assignments in hospitals (Fei et al., 2008) and workforce scheduling (Beliën and Demeulemeester, 2007; Everborn and Ronnqvist, 2004; Eveborn et al., 2006). Because of the extremely broad range of applicability (as highlighted above), the SPP can be interpreted from a wide variety of perspectives. ...
In this paper, we present a linear programming (LP) model of the set partitioning problem (SPP). The number of variables and the number of constraints of the proposed model are bounded by (third-degree) polynomial functions of the number of non-zero entries of the SPP input matrix, respectively. Hence, the model provides a new affirmative resolution to the all-important 'P vs. NP' question. We use a transportation problem-based reformulation that we develop, and a path-based modelling approach similar to that used in Diaby (2007) to formulate the proposed LP model. The approach is illustrated with a numerical example.
... The task is to allocate clients on (possibly multiple) rings, satisfying the bandwidth constraints on the rings and minimizing the equipment cost. This problem has been tackled previously using mixed integer programming (MIP)[1] and constraint programming (CP)[2, 3]. Much attention was devoted to variable branching heuristics and breaking ring symmetries (since all rings are identical). ...
... Please refer to Section 5 in [2] for a detailed discussion. Another CP model was proposed in [3] and it broke symmetries by adding a lexicographic bound to set variable domain. With the additional lexicographic component, the solver obtained a tighter approximation of the set-variable domain . ...
... Smith's algorithm [2] used ILOG Solver on one 1.7GHz processor . Hybrid[3] was run using the Eclipse constraint solver on a Pentium 4 2GHz processor, with a timeout of 3000 seconds.Table 2 reports the CPU time and number of backtracks (bt) required for each approach to prove the optimality of each instance . Our Boosting algorithm is, on average, more than 3700 times faster than the MIP and Hybrid approaches and visits several orders on magnitude less nodes than them. ...
This paper reconsiders the deployment of synchronous op- tical networks (SONET), an optimization problem naturally expressed in terms of set variables. Earlier approaches, using either MIP or CP technologies, focused on symmetry breaking, including the use of SBDS, and the design of eective branching strategies. This paper advocates an orthogonal approach and argues that the thrashing behavior experi- enced in earlier attempts is primarily due to a lack of pruning. It studies how to improve domain ltering by taking a more global view of the application and imposing redundant global constraints. The technical results include novel hardness results, propagation algorithms for global constraints, and inference rules. The paper also evaluates the contribu- tions experimentally by presenting a novel model with static symmetric- breaking constraints and a static variable ordering which is many orders of magnitude faster than existing approaches.
... The present work is primarily on the benefits of nonzero λ, and so a detailed study of general set programming techniques would be inappropriate; but we made informal tests of several other set-programming solvers. We had hoped that a solver using containment-lexicographic hybrid bounds as described by Sadler and Gervet (Sadler and Gervet, 2008) would offer good performance, and chose the ECL i PS e framework partly to gain access to its ic hybrid sets implementation of such bounds. In practice, however, ic hybrid sets gave consistently worse performance than ic sets (typically by an approximate factor of two). ...
Constructing an encoding of a concept lattice using short bit vectors allows for efficient computation of join operations on the lattice. Join is the central operation any unification-based parser must support. We extend the traditional bit vector encoding, which represents join failure using the zero vector, to count any vector with less than a fixed number of one bits as failure. This allows non-joinable elements to share bits, resulting in a smaller vector size. A constraint solver is used to construct the encoding, and a variety of techniques are employed to find near-optimal solutions and handle timeouts. An evaluation is provided comparing the extended representation of failure with traditional bit vector techniques.
... The addition of a cardinality component to the subset-bound representation, and the associated pruning rules, was proposed in [3, 4]. This richer domain was further enhanced by a lexicographic component in [5, 6] since many set constraint satisfaction problems contain many symmetries which can broken by lexicographic constraints. 0/1 matrix models can also be used to encode the characteristic function of the subset-bound domain, while cardinality and lexicographic restrictions are expressed by posting constraints [7, 8]. ...
This paper presents the rst experimental evaluation of the length-lex domain for set variables. The implementation is based on bound-consistency algorithms proposed in earlier work and two novel technical contributions: a generic ltering algorithm which automatically pushes ordering constraints into symmetric binary constraints with only a logarithmic overhead and an adaptation of symmetry-breaking con- straints from 0/1 matrices to the length-lex ordering. The experimental results indicate that the length-lex representation for set variables is very eective and robust on traditional set-CSPs benchmarks.
