269 research outputs found
A Logical Approach to Efficient Max-SAT solving
Weighted Max-SAT is the optimization version of SAT and many important
problems can be naturally encoded as such. Solving weighted Max-SAT is an
important problem from both a theoretical and a practical point of view. In
recent years, there has been considerable interest in finding efficient solving
techniques. Most of this work focus on the computation of good quality lower
bounds to be used within a branch and bound DPLL-like algorithm. Most often,
these lower bounds are described in a procedural way. Because of that, it is
difficult to realize the {\em logic} that is behind.
In this paper we introduce an original framework for Max-SAT that stresses
the parallelism with classical SAT. Then, we extend the two basic SAT solving
techniques: {\em search} and {\em inference}. We show that many algorithmic
{\em tricks} used in state-of-the-art Max-SAT solvers are easily expressable in
{\em logic} terms with our framework in a unified manner.
Besides, we introduce an original search algorithm that performs a restricted
amount of {\em weighted resolution} at each visited node. We empirically
compare our algorithm with a variety of solving alternatives on several
benchmarks. Our experiments, which constitute to the best of our knowledge the
most comprehensive Max-sat evaluation ever reported, show that our algorithm is
generally orders of magnitude faster than any competitor
Tractability in Constraint Satisfaction Problems: A Survey
International audienceEven though the Constraint Satisfaction Problem (CSP) is NP-complete, many tractable classes of CSP instances have been identified. After discussing different forms and uses of tractability, we describe some landmark tractable classes and survey recent theoretical results. Although we concentrate on the classical CSP, we also cover its important extensions to infinite domains and optimisation, as well as #CSP and QCSP
The complexity of Boolean surjective general-valued CSPs
Valued constraint satisfaction problems (VCSPs) are discrete optimisation
problems with a -valued objective function given as
a sum of fixed-arity functions. In Boolean surjective VCSPs, variables take on
labels from and an optimal assignment is required to use both
labels from . Examples include the classical global Min-Cut problem in
graphs and the Minimum Distance problem studied in coding theory.
We establish a dichotomy theorem and thus give a complete complexity
classification of Boolean surjective VCSPs with respect to exact solvability.
Our work generalises the dichotomy for -valued constraint
languages (corresponding to surjective decision CSPs) obtained by Creignou and
H\'ebrard. For the maximisation problem of -valued
surjective VCSPs, we also establish a dichotomy theorem with respect to
approximability.
Unlike in the case of Boolean surjective (decision) CSPs, there appears a
novel tractable class of languages that is trivial in the non-surjective
setting. This newly discovered tractable class has an interesting mathematical
structure related to downsets and upsets. Our main contribution is identifying
this class and proving that it lies on the borderline of tractability. A
crucial part of our proof is a polynomial-time algorithm for enumerating all
near-optimal solutions to a generalised Min-Cut problem, which might be of
independent interest.Comment: v5: small corrections and improved presentatio
An Analysis of Core-Guided Maximum Satisfiability Solvers Using Linear Programming
Many current complete MaxSAT algorithms fall into two categories: core-guided or implicit hitting set. The two kinds of algorithms seem to have complementary strengths in practice, so that each kind of solver is better able to handle different families of instances. This suggests that a hybrid might match and outperform either, but the techniques used seem incompatible. In this paper, we focus on PMRES and OLL, two core-guided algorithms based on max resolution and soft cardinality constraints, respectively. We show that these algorithms implicitly discover cores of the original formula, as has been previously shown for PM1. Moreover, we show that in some cases, including unweighted instances, they compute the optimum hitting set of these cores at each iteration. We also give compact integer linear programs for each which encode this hitting set problem. Importantly, their continuous relaxation has an optimum that matches the bound computed by the respective algorithms. This goes some way towards resolving the incompatibility of implicit hitting set and core-guided algorithms, since solvers based on the implicit hitting set algorithm typically solve the problem by encoding it as a linear program
Higher-Level Consistencies: Where, When, and How Much
Determining whether or not a Constraint Satisfaction Problem (CSP) has a solution is NP-complete. CSPs are solved by inference (i.e., enforcing consistency), conditioning (i.e., doing search), or, more commonly, by interleaving the two mechanisms. The most common consistency property enforced during search is Generalized Arc Consistency (GAC). In recent years, new algorithms that enforce consistency properties stronger than GAC have been proposed and shown to be necessary to solve difficult problem instances.
We frame the question of balancing the cost and the pruning effectiveness of consistency algorithms as the question of determining where, when, and how much of a higher-level consistency to enforce during search. To answer the `where\u27 question, we exploit the topological structure of a problem instance and target high-level consistency where cycle structures appear. To answer the \u27when\u27 question, we propose a simple, reactive, and effective strategy that monitors the performance of backtrack search and triggers a higher-level consistency as search thrashes. Lastly, for the question of `how much,\u27 we monitor the amount of updates caused by propagation and interrupt the process before it reaches a fixpoint. Empirical evaluations on benchmark problems demonstrate the effectiveness of our strategies.
Adviser: B.Y. Choueiry and C. Bessier
Optimization and inference under fuzzy numerical constraints
Εκτεταμένη έρευνα έχει γίνει στους τομείς της Ικανοποίησης Περιορισμών με
διακριτά (ακέραια) ή πραγματικά πεδία τιμών. Αυτή η έρευνα έχει οδηγήσει σε
πολλαπλές σημασιολογικές περιγραφές, πλατφόρμες και
συστήματα για την περιγραφή σχετικών προβλημάτων με επαρκείς βελτιστοποιήσεις.
Παρά ταύτα, λόγω της ασαφούς φύσης
πραγματικών προβλημάτων ή ελλιπούς μας γνώσης για αυτά, η σαφής μοντελοποίηση
ενός προβλήματος ικανοποίησης περιορισμών δεν είναι πάντα ένα εύκολο ζήτημα ή
ακόμα και η καλύτερη προσέγγιση. Επιπλέον, το πρόβλημα της μοντελοποίησης και
επίλυσης ελλιπούς γνώσης είναι ακόμη δυσκολότερο. Επιπροσθέτως, πρακτικές
απαιτήσεις μοντελοποίησης και μέθοδοι βελτιστοποίησης του χρόνου αναζήτησης
απαιτούν συνήθως ειδικές πληροφορίες για το πεδίο εφαρμογής,
καθιστώντας τη δημιουργία ενός γενικότερου πλαισίου βελτιστοποίησης ένα
ιδιαίτερα δύσκολο πρόβλημα. Στα πλαίσια αυτής της εργασίας θα μελετήσουμε το
πρόβλημα της μοντελοποίησης και αξιοποίησης σαφών, ελλιπών ή ασαφών
περιορισμών, καθώς και πιθανές στρατηγικές βελτιστοποίησης. Καθώς τα
παραδοσιακά προβλήματα ικανοποίησης περιορισμών λειτουργούν βάσει συγκεκριμένων
και προκαθορισμένων κανόνων και σχέσεων, παρουσιάζει ενδιαφέρον η διερεύνηση
στρατηγικών και βελτιστοποιήσεων που θα επιτρέπουν το συμπερασμό νέων ή/και
αποδοτικότερων περιορισμών. Τέτοιοι επιπρόσθετοι κανόνες θα μπορούσαν να
βελτιώσουν τη διαδικασία αναζήτησης μέσω της εφαρμογής αυστηρότερων περιορισμών
και περιορισμού του χώρου αναζήτησης ή να προσφέρουν χρήσιμες πληροφορίες στον
αναλυτή για τη φύση του προβλήματος που
μοντελοποιεί.Extensive research has been done in the areas of Constraint Satisfaction with
discrete/integer
and real domain ranges. Multiple platforms and systems to deal with these kinds
of domains have been developed and appropriately optimized. Nevertheless, due
to the incomplete and possibly vague nature of real-life problems, modeling a
crisp and adequately strict satisfaction problem may not always be easy or even
appropriate. The problem of modeling incomplete
knowledge or solving an incomplete/relaxed representation of a problem is a
much harder issue to tackle. Additionally, practical modeling requirements and
search optimizations require specific domain knowledge in order to be
implemented, making the creation of a more generic optimization framework an
even harder problem.In this thesis, we will study the problem of modeling and
utilizing incomplete and fuzzy constraints, as well as possible optimization
strategies. As constraint satisfaction problems usually contain hard-coded
constraints based on specific problem and domain knowledge, we will investigate
whether strategies and generic heuristics exist for inferring new constraint
rules. Additional rules could optimize the search process by implementing
stricter constraints and thus pruning the search space or even provide useful
insight to the researcher concerning the nature of the investigated problem
Higher-Level Consistencies: Where, When, and How Much
Determining whether or not a Constraint Satisfaction Problem (CSP) has a solution is NP-complete. CSPs are solved by inference (i.e., enforcing consistency), conditioning (i.e., doing search), or, more commonly, by interleaving the two mechanisms. The most common consistency property enforced during search is Generalized Arc Consistency (GAC). In recent years, new algorithms that enforce consistency properties stronger than GAC have been proposed and shown to be necessary to solve difficult problem instances.
We frame the question of balancing the cost and the pruning effectiveness of consistency algorithms as the question of determining where, when, and how much of a higher-level consistency to enforce during search. To answer the `where\u27 question, we exploit the topological structure of a problem instance and target high-level consistency where cycle structures appear. To answer the \u27when\u27 question, we propose a simple, reactive, and effective strategy that monitors the performance of backtrack search and triggers a higher-level consistency as search thrashes. Lastly, for the question of `how much,\u27 we monitor the amount of updates caused by propagation and interrupt the process before it reaches a fixpoint. Empirical evaluations on benchmark problems demonstrate the effectiveness of our strategies.
Adviser: B.Y. Choueiry and C. Bessier
Higher-Level Consistencies: Where, When, and How Much
Determining whether or not a Constraint Satisfaction Problem (CSP) has a solution is NP-complete. CSPs are solved by inference (i.e., enforcing consistency), conditioning (i.e., doing search), or, more commonly, by interleaving the two mechanisms. The most common consistency property enforced during search is Generalized Arc Consistency (GAC). In recent years, new algorithms that enforce consistency properties stronger than GAC have been proposed and shown to be necessary to solve difficult problem instances.
We frame the question of balancing the cost and the pruning effectiveness of consistency algorithms as the question of determining where, when, and how much of a higher-level consistency to enforce during search. To answer the `where\u27 question, we exploit the topological structure of a problem instance and target high-level consistency where cycle structures appear. To answer the \u27when\u27 question, we propose a simple, reactive, and effective strategy that monitors the performance of backtrack search and triggers a higher-level consistency as search thrashes. Lastly, for the question of `how much,\u27 we monitor the amount of updates caused by propagation and interrupt the process before it reaches a fixpoint. Empirical evaluations on benchmark problems demonstrate the effectiveness of our strategies.
Adviser: B.Y. Choueiry and C. Bessier
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