Experimental Evaluation of Branching Schemes for the CSP

The search strategy of a CP solver is determined by the variable and value ordering heuristics it employs and by the branching scheme it follows. Although the effects of variable and value ordering heuristics on search effort have been widely studied, the effects of different branching schemes have received less attention. In this paper we study this effect through an experimental evaluation that includes standard branching schemes such as 2-way, d-way, and dichotomic domain splitting, as well as variations of set branching where branching is performed on sets of values. We also propose and evaluate a generic approach to set branching where the partition of a domain into sets is created using the scores assigned to values by a value ordering heuristic, and a clustering algorithm from machine learning. Experimental results demonstrate that although exponential differences between branching schemes, as predicted in theory between 2-way and d-way branching, are not very common, still the choice of branching scheme can make quite a difference on certain classes of problems. Set branching methods are very competitive with 2-way branching and outperform it on some problem classes. A statistical analysis of the results reveals that our generic clustering-based set branching method is the best among the methods compared.Comment: To appear in the 3rd workshop on techniques for implementing constraint programming systems (TRICS workshop at the 16th CP Conference), St. Andrews, Scotland 201

Collective Singleton-Based Consistency for Qualitative Constraint Networks

Partial singleton closure under weak composition, or partial singleton (weak) path-consistency for short, is essential for approximating satisfiability of qualitative constraints networks. Briefly put, partial singleton path-consistency ensures that each base relation of each of the constraints of a qualitative constraint network can define a singleton relation in the corresponding partial closure of that network under weak composition, or in its corresponding partially (weak) path-consistent subnetwork for short. In particular, partial singleton path-consistency has been shown to play a crucial role in tackling the minimal labeling problem of a qualitative constraint network, which is the problem of finding the strongest implied constraints of that network. In this paper, we propose a stronger local consistency that couples partial singleton path-consistency with the idea of collectively deleting certain unfeasible base relations by exploiting singleton checks. We then propose an efficient algorithm for enforcing this consistency that, given a qualitative constraint network, performs fewer constraint checks than the respective algorithm for enforcing partial singleton path-consistency in that network. We formally prove certain properties of our new local consistency, and motivate its usefulness through demonstrative examples and a preliminary experimental evaluation with qualitative constraint networks of Interval Algebra

On neighbourhood singleton-style consistencies for qualitative spatial and temporal reasoning

Given a qualitative constraint network (QCN), a singleton-style consistency focuses on each base relation (atom) of a constraint separately, rather than the entire constraint altogether. This local consistency is essential for tackling fundamental reasoning problems associated with QCNs, such as minimal labeling, but can suffer from redundant constraint checks, especially when checks occur far from where the pruning usually takes place. In this paper, we propose singleton-style consistencies that are applied just on the neighbourhood of a singleton-checked constraint instead of the whole network. We make a theoretical comparison with existing consistencies and consequently prove some properties of the new ones. Further, we propose algorithms to enforce our consistencies, as well as parsimonious variants thereof, that are more efficient in practice than the state of the art. An experimental evaluation with random and structured QCNs of Allen's Interval Algebra in the phase transition region demonstrates the potential of our approach.acceptedVersionPeer reviewe

Multi-Armed Bandits for Adaptive Constraint Propagation

International audienceAdaptive constraint propagation has recently received a great attention. It allows a constraint solver to exploit various levels of propagation during search, and in many cases it shows better performance than static/predefined. The crucial point is to make adaptive constraint propagation automatic , so that no expert knowledge or parameter specification is required. In this work, we propose a simple learning technique, based on multi-armed bandits, that allows to automatically select among several levels of propagation during search. Our technique enables the combination of any number of levels of propagation whereas existing techniques are only defined for pairs. An experimental evaluation demonstrates that the proposed technique results in a more efficient and stable solver

On Neighborhood Singleton Consistencies

International audienceCP solvers predominantly use arc consistency (AC) as the default propagation method. Many stronger consistencies, such as triangle consistencies (e.g. RPC and maxRPC) exist, but their use is limited despite results showing that they outperform AC on many problems. This is due to the intricacies involved in incorporating them into solvers. On the other hand, singleton consistencies such as SAC can be easily crafted into solvers but they are too expensive. We seek a balance between the efficiency of triangle consistencies and the ease of implementation of singleton ones. Using the recently proposed variant of SAC called Neighborhood SAC as basis, we propose a family of weaker singleton consistencies. We study them theoretically, comparing their pruning power to existing consistencies. We make a detailed experimental study using a very simple algorithm for their implementation. Results demonstrate that they outperform the existing propagation techniques, often by orders of magnitude, on a wide range of problems

Αποδοτικοί αλγόριθμοι ισχυρής συνέπειας και προσαρμοστικές τεχνικές για προβλήματα ικανοποίησης περιορισμών

Constraint Programming (CP) is a successful technology for solving a wide range of problemsin business and industry which require the satisfaction of a set of complex constraints.Examples include product configuration, resource allocation, transportation, andscheduling. As the simultaneous satisfaction of different constraints is intractable in general,problems can become very difficult to solve as their size increases. CP has thusdeveloped various techniques to tackle this inherent problem. Enforcing a local consistencyproperty is one of the most important such techniques.Bounds Consistency (BC) and Generalised Arc Consistency (GAC) are the two mostwidely studied and used local consistencies in CP solvers. While there exist strongerlocal consistency (SLC) properties, their usage is limited due to their prohibitive cost.Examples are max Restricted Path Consistency (maxRPC) and max Restricted PairWiseConsistency (maxRPWC).In our research, we propose efficient filtering algorithms for enforcing SLCs. In particular,we propose new algorithms for maxRPC and maxRPWC that advance the existingalgorithms (theoretically and practically). We also propose algorithms that achieve weakerconsistencies with a lower cost. In addition, we have extended the recent algorithms fromthe family of Simple Tabular Reduction (STR) to achieve a higher-order local consistencyproperty. Experiments demonstrate that these algorithms can significantly outperformvarious state-ot the-art (G)AC algorithms, even by orders of magnitude, and thus can becomevery useful additions to the propagation techniques that CP solvers currently apply.Additionally, we have introduced and defined a new strong Bounds Consistency, calledPWBC, as well as a polynomial filtering algorithm based on this consistency for the importantclass of linear inequalities. Theoretical and experimental results demonstrate thepotential of SLCs that reason on bounds.Finally, since SLCs may still be too expensive to maintain during search in manyproblems, we have suggested ways to interleave them with weaker propagation methodssuch as GAC. We have proposed fully automated heuristics that can dynamically selectthe most appropriate filtering algorithm. All algorithms are incorporated in an adaptivefiltering scheme to further tackle the inherent difficulty of constraint satisfaction, resultingin a more robust constraint solver. Overall, this research proposes filtering algorithms andadaptive techniques that exploit the filtering power offered by SLCs in an efficient way, inorder to increase the efficacy of CP solvers.Ο Προγραμματισμός με Περιορισμούς (Constraint Programming - CP) είναι μια επιτυχημένητεχνολογία για την επίλυση πολλών προβλημάτων από το χώρο των επιχειρήσεων και τηςβιομηχανίας, που απαιτούν την ικανοποίηση μιας σειράς πολύπλοκων περιορισμών.Παραδείγματα τέτοιων προβλημάτων είναι η διαμόρφωση προϊόντος, η κατανομή πόρων, ταπροβλήματα μεταφοράς και χρονοπρογραμματισμού. Επειδή η ταυτόχρονη ικανοποίηση τωνδιαφόρων περιορισμών είναι γενικά δυσεπίλυτη, τα προβλήματα μπορεί να γίνουν ακόμηδυσκολότερα καθώς αυξάνει το μέγεθός τους. Ο Προγραμματισμός με Περιορισμούς έχειαναπτύξει διάφορες τεχνικές για να αντιμετωπίσει αυτό το εγγενές πρόβλημα. Μια από τις πιοσημαντικές τέτοιες τεχνικές είναι η εφαρμογή τοπικής συνέπειας.Οι τοπικές συνέπειες που έχουν ευρέως μελετηθεί και χρησιμοποιηθεί από συστήματαεπίλυσης είναι η συνέπεια ορίων (Bounds Consistency - BC) και η συνέπεια τόξου (ArcConsistency - AC). Παρότι έχουν προταθεί και ισχυρότερες τοπικές συνέπειες, η χρήση τους είναιπεριορισμένη λόγω του απαγορευτικού κόστους τους. Παραδείγματα αποτελούν οι συνέπειες maxRestricted Path Consistency (maxRPC) και max Restricted PairWise Consistency (maxRPWC).Στην παρούσα έρευνα προτείνουμε αποδοτικούς αλγόριθμους ελέγχου συνέπειας για τηνεπιβολή ισχυρών τοπικών συνεπειών. Συγκεκριμένα, προτείνουμε νέους αλγόριθμους για τιςσυνέπειες maxRPC και maxRPWC που βελτιώνουν (θεωρητικά και πρακτικά) τουςπροηγούμενους. Επίσης, προτείνουμε αλγόριθμους που εφαρμόζουν ασθενέστερες συνέπειες μεχαμηλότερο κόστος, έχουμε επεκτείνει τους πρόσφατους από την οικογένεια των STR (SimpleTabular Reduction) αλγορίθμων για την επίτευξη υψηλότερης τάξης (higher-order) τοπικήςσυνέπειας. Πειράματα δείχνουν ότι αυτοί οι αλγόριθμοι μπορούν να ξεπεράσουν state-ot-the-artAC αλγόριθμους με σημαντικές διαφορές, ακόμη και κατά τάξεις μεγέθους, και συνεπώς, μπορούννα αποτελέσουν χρήσιμες προσθήκες στις τεχνικές διάδοσης περιορισμών για τους σύγχρονουςCP επιλυτές. Επιπρόσθετα, εισάγουμε και ορίζουμε μια νέα ισχυρή συνέπεια ορίων, πουονομάζεται PWBC, καθώς και έναν πολυωνυμικό αλγόριθμο ελέγχου συνέπειας που βασίζεται σεαυτήν για την σημαντική κατηγορία των γραμμικών ανισοτήτων. Τα θεωρητικά και πειραματικάαποτελέσματα αναδεικνύουν τις δυνατότητες των ισχυρών συνεπειών που επιβάλλονται στα όρια.Τέλος, δεδομένου ότι οι ισχυρές συνέπειες μπορεί να εξακολουθούν να είναι ακριβές στηνεφαρμογή τους κατά την αναζήτηση σε πολλά προβλήματα, προτείνουμε τρόπους ώστε ναπαρεμβάλλονται μαζί με ασθενέστερες συνέπειες, όπως η συνέπεια τόξου. Προτείνουμε πλήρωςαυτοματοποιημένες ευρετικές μεθόδους, που μπορούν να επιλέξουν δυναμικά τον καταλληλότεροαλγόριθμο φιλτραρίσματος. Οι προτεινόμενοι αλγόριθμοι ενσωματώνονται σε ένα προσαρμοστικόσύστημα διάδοσης περιορισμών για να αντιμετωπίσει περαιτέρω την εγγενή δυσκολίαικανοποίησης περιορισμών, με αποτέλεσμα έναν ισχυρότερο επιλυτή. Συνολικά, η έρευναπροτείνει αλγόριθμους φιλτραρίσματος και προσαρμοστικές τεχνικές που εκμεταλλεύονται τομεγάλο φιλτράρισμα των ισχυρών συνεπειών με αποτελεσματικό τρόπο, προκειμένου να αυξηθείη αποτελεσματικότητα των CP επιλυτών

A Lazy Algorithm to Efficiently Approximate Singleton Path Consistency for Qualitative Constraint Networks

International audienc

On the utility of neighbourhood singleton-style consistencies for qualitative constraint-based spatial and temporal reasoning

A singleton-style consistency is a local consistency that verifies if each base relation (atom) of each constraint of a qualitative constraint network (QCN) can serve as a support with respect to the closure of that network under a (naturally) weaker local consistency. This local consistency is essential for tackling fundamental reasoning problems associated with QCNs, such as the satisfiability checking or the minimal labeling problem, but can suffer from redundant constraint checks, especially when those checks occur far from where the pruning usually takes place. In this paper, we propose singleton-style consistencies that are applied just on the neighbourhood of a singleton-checked constraint instead of the whole network. We make a theoretical comparison with existing consistencies and consequently prove some properties of the new ones. In addition, we propose algorithms to enforce our consistencies, as well as parsimonious variants thereof, that are more efficient in practice than the state of the art. We make an experimental evaluation with random and structured QCNs of Interval Algebra in the phase transition region to demonstrate the potential of our approach.Peer reviewe

Extending STR to a Higher-Order Consistency

International audienceOne of the most widely studied classes of constraints in constraint programming (CP) is that of table constraints. Numerous specialized filtering algorithms, enforcing the wellknown property called generalized arc consistency (GAC), have been developed for such constraints. Among the most successful GAC algorithms for table constraints, we find variants of simple tabular reduction (STR), like STR2. In this paper, we propose an extension of STR-based algorithms that achieves full pairwise consistency (FPWC), a consistency stronger than GAC and max restricted pairwise consistency (maxRPWC). Our approach involves counting the number of occurrences of specific combinations of values in constraint intersections. Importantly, the worst-case time complexity of one call to the basic filtering procedure at the heart of our new algorithm is quite close to that of STR algorithms. Experiments demonstrate that our method can outperform STR2 in many classes of problems, being significantly faster in some cases. Also, it is clearly superior to maxRPWC+, an algorithm that has been recently proposed