On Path Consistency for Binary Constraint Satisfaction Problems

Constraint satisfaction problems (CSPs) provide a flexible and powerful framework for modeling and solving many decision problems of practical importance. Consistency properties and the algorithms for enforcing them on a problem instance are at the heart of Constraint Processing and best distinguish this area from other areas concerned with the same combinatorial problems. In this thesis, we study path consistency (PC) and investigate several algorithms for enforcing it on binary finite CSPs. We also study algorithms for enforcing consistency properties that are related to PC but are stronger or weaker than PC. We identify and correct errors in the literature and settle an open question. We propose two improvements that we apply to the well-known algorithms PC-8 and PC-2001, yielding PC-8+ and PC-2001+. Further, we propose a new algorithm for enforcing partial path consistency, σ-∆-PPC, which generalizes features of the well-known algorithms DPC and PPC. We evaluate over fifteen different algorithms on both benchmark and randomly generated binary problems to empirically demonstrate the effectiveness of our approach. Adviser: Berthe Y. Choueir

On tree-preserving constraints

© 2017, Springer International Publishing Switzerland. The study of tractable subclasses of constraint satisfaction problems is a central topic in constraint solving. Tree convex constraints are extensions of the well-known row convex constraints. Just like the latter, every path-consistent tree convex constraint network is globally consistent. However, it is NP-complete to decide whether a tree convex constraint network has solutions. This paper studies and compares three subclasses of tree convex constraints, which are called chain-, path-, and tree-preserving constraints respectively. The class of tree-preserving constraints strictly contains the subclasses of path-preserving and arc-consistent chain-preserving constraints. We prove that, when enforcing strong path-consistency on a tree-preserving constraint network, in each step, the network remains tree-preserving. This ensures the global consistency of consistent tree-preserving networks after enforcing strong path-consistency, and also guarantees the applicability of the partial path-consistency algorithms to tree-preserving constraint networks, which is usually much more efficient than the path-consistency algorithms for large sparse constraint networks. As an application, we show that the class of tree-preserving constraints is useful in solving the scene labelling problem

A Framework for Decision-based Consistencies

International audienceConsistencies are properties of constraint networks that can be enforced by appropriate algorithms to reduce the size of the search space to be explored. Recently, many consistencies built upon taking decisions (most often, variable assignments) and stronger than (general- ized) arc consistency have been introduced. In this paper, our ambition is to present a clear picture of decision-based consistencies. We identify four general classes (or levels) of decision-based consistencies, denoted by S∆φ, E∆φ, B∆φ and D∆φ, study their relationships, and show that known consistencies are particular cases of these classes. Interestingly, this gen- eral framework provides us with a better insight into decision-based con- sistencies, and allows us to derive many new consistencies that can be directly integrated and compared with other ones

STR2: Optimized Simple Tabular Reduction for Table Constraints

International audienceTable constraints play an important role within constraint programming. Recently, many schemes or algorithms have been proposed to propagate table constraints and/or to compress their representation. In this paper, we describe an optimization of simple tabular reduction (STR), a technique proposed by J. Ullmann to dynamically maintain the tables of supports when generalized arc consistency (GAC) is enforced/maintained. STR2, the new refined GAC algorithm we propose, allows us to limit the number of operations related to validity checking and search of supports. Interestingly enough, this optimization makes simple tabular reduction potentially r times faster where r is the arity of the constraint(s). The results of an extensive experimentation that we have conducted with respect to random and structured instances indicate that STR2 is usually around twice as fast as the original STR, two or three times faster than the approach based on the hidden variable encoding, and can be up to one order of magnitude faster than previously state-of-the art (generic) GAC algorithms on some series of instances. When comparing STR2 with the more recently developed algorithm based on multi-valued decision diagrams (MDDs), we show that both approaches are rather complementary

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

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

Conservative Dual Consistency

Consistencies are properties of Constraint Networks (CNs) that can be exploited in order to make inferences. When a significant amount of such inferences can be performed, CNs are much easier to solve. In this paper, we interest ourselves in relation filtering consistencies for binary constraints, i.e. consistencies that allow to identify inconsistent pairs of values. We propose a new consistency called Dual Consistency (DC) and relate it to Path Consistency (PC). We show that Conservative DC (CDC, i.e. DC with only relations associated with the constraints of the network considered) is more powerful, in terms of filtering, than Conservative PC (CPC). Following the approach of Mac Gregor, we introduce an algorithm to establish (strong) CDC with a very low worst-case space complexity. Even if the relative efficiency of the algorithm introduced to establish (strong) CDC partly depends on the density of the constraint graph, the experiments we have conducted show that, on many series of CSP instances, CDC is largely faster than CPC (up to more than one order of magnitude). Besides, we have observed that enforcing CDC in a preprocessing stage can significantly speed up the resolution of hard structured instances

Second-Order Consistencies

International audienceIn this paper, we propose a comprehensive study of second-order consistencies (i.e., consistencies identifying inconsistent pairs of values) for constraint satisfaction. We build a full picture of the relationships existing between four basic second-order consistencies, namely path consistency (PC), 3-consistency (3C), dual consistency (DC) and 2-singleton arc consistency (2SAC), as well as their conservative and strong variants. Interestingly, dual consistency is an original property that can be established by using the outcome of the enforcement of generalized arc consistency (GAC), which makes it rather easy to obtain since constraint solvers typically maintain GAC during search. On binary constraint networks, DC is equivalent to PC, but its restriction to existing constraints, called conservative dual consistency (CDC), is strictly stronger than traditional conservative consistencies derived from path consistency, namely partial path consistency (PPC) and conservative path consistency (CPC). After introducing a general algorithm to enforce strong (C) DC, we present the results of an experimentation over a wide range of benchmarks that demonstrate the interest of (conservative) dual consistency. In particular, we show that enforcing (C) DC before search clearly improves the performance of MAC (the algorithm that maintains GAC during search) on several binary and non-binary structured problems