Restarts are an important technique to make search more robust. This paper is concerned with how to maintain and propagate nogoods recorded from restarts efficiently. It builds on reduced nld-nogoods introduced for restarts and increasing nogoods introduced for symmetry breaking. The paper shows that reduced nld-nogoods extracted from a single restart are in fact increasing, which can thus benefit from the efficient propagation algorithm of the incNGs global constraint. We present a lighter weight filtering algorithm for incNGs in the context of restart-based search using dynamic event sets (dynamic subscriptions). We show formally that the lightweight version enforces GAC on each nogood while reducing the number of subscribed decisions. The paper also introduces an efficient approximation to nogood minimization such that all shortened reduced nld-nogoods from the same restart are also increasing and can be propagated with the new filtering algorithm. Experimental results confirm that our lightweight filtering algorithm and approximated nogood minimization successfully trade a slight loss in pruning for considerably better efficiency, and hence compare favorably against existing state-of-the-art techniques.

Constraint Programming (CP) solvers classically explore the solution space using tree-search based heuristics. Monte-Carlo Tree-Search (MCTS) is a tree-search method aimed at optimal sequential decision making under uncertainty. At the crossroads of CP and MCTS, this paper presents the Bandit Search for Constraint Programming (BASCOP)  algorithm, adapting MCTS to the specifics of CP search trees. Formally, MCTS simultaneously estimates the average node reward, and uses it to bias the exploration towards the most promising regions of the tree, borrowing the multi-armed bandit (MAB) decision rule. The two contributions in BASCOP concern i) a specific reward function, estimating the relative failure depth conditionally to a (variable, value) assignment; ii) a new  decision rule, hybridizing the MAB framework and the spirit of local neighborhood search. Specifically, BASCOP guides the CP search in the neighborhood of the previous best solution, by exploiting statistical estimates gathered across multiple restarts. BASCOP, using Gecode as the underlying constraint solver, shows significant improvements over the depth-first search baseline on some  CP benchmark suites. For hard job-shop scheduling problems, BASCOP matches the results of state-of-the-art scheduling-specific CP approaches. These results demonstrate the potential of BASCOP as a generic yet robust search method for CP.

When implementing a propagator for a constraint, one must decide about variants: When implementing min, should one also implement max? Should one implement linear constraints both with unit and non-unit coefficients? Constraint variants are ubiquitous: implementing them requires considerable (if not prohibitive) effort and decreases maintainability, but will deliver better performance than resorting to constraint decomposition. This paper shows how to use views to derive perfect propagator variants. A model for views and derived propagators is introduced. Derived propagators are proved to be indeed perfect in that they inherit essential properties such as correctness and domain and bounds consistency. Techniques for systematically deriving propagators such as transformation, generalization, specialization, and type conversion are developed. The paper introduces an implementation architecture for views that is independent of the underlying constraint programming system. A detailed evaluation of views implemented in Gecode shows that derived propagators are efficient and that views often incur no overhead. Without views, Gecode would either require 180 000 rather than 40 000 lines of propagator code, or would lack many efficient propagator variants. Compared to 8 000 lines of code for views, the reduction in code for propagators yields a 1750% return on investment.

When implementing a propagator for a constraint, one must decide about variants: When implementing min, should one also implement max? Should one implement linear equations both with and without coefficients? Constraint variants are ubiquitous: implementing them requires considerable (if not prohibitive) effort and decreases maintainability, but will deliver better performance. This paper shows how to use variable views, previously introduced for an implementation architecture, to derive perfect propagator variants. A model for views and derived propagators is introduced. Derived propagators are proved to be indeed perfect in that they inherit essential properties such as correctness and domain and bounds consistency. Techniques for systematically deriving propagators such as transformation, generalization, specialization, and channeling are developed for several variable domains. We evaluate the massive impact of derived propagators. Without derived propagators, Gecode would require 140000 rather than 40000 lines of code for propagators.

This paper presents a model and implementation techniques for speeding up constraint propagation. Three fundamental approaches to improving constraint propagation based on propagators as implementations of constraints are explored: keeping track of which propagators are at fixpoint, choosing which propagator to apply next, and how to combine several propagators for the same constraint. We show how idempotence reasoning and events help track fixpoints more accurately. We improve these methods by using them dynamically (taking into account current domains to improve accuracy). We define priority-based approaches to choosing a next propagator and show that dynamic priorities can improve propagation. We illustrate that the use of multiple propagators for the same constraint can be advantageous with priorities, and introduce staged propagators that combine the effects of multiple propagators with priorities for greater efficiency.