Adaptive Parameterized Consistency

Abstract

Abstract. State-of-the-art constraint solvers uniformly maintain the same level of local consistency (usually arc consistency) on all the instances. We propose parameterized local consistency, an original approach to adjust the level of consistency depending on the instance and on which part of the instance we propagate. We do not use as parameter one of the features of the instance, as done for instance in portfolios of solvers. We use as parameter the stability of values, which is a feature based on the state of the arc consistency algorithm during its execution. Parameterized local consistencies choose to enforce arc consistency or a higher level of local consistency on a value depending on whether the stability of the value is above or below a given threshold. We also propose a way to dynamically adapt the parameter, and thus the level of local consistency, during search. This approach allows us to get a good trade-off between the number of values pruned and the computational cost. We validate our approach on various problems from the CSP competition.

Adaptive filtering strategy for numerical constraint satisfaction problems

“…Once the level is learned, it is statically applied during the entire search. In Balafrej, Bessiere, Coletta, and Bouyakhf (2013), for each variable/constraint, the solver learns during the search a parameter that characterizes a parameterized level of consistency to apply to the variable/constraint. That parameterized level lies between AC and a stronger level.…”

Adaptive Parameterized Consistency for Non-binary CSPs by Counting Supports

“…Figure 1 illustrates an example for the constraint x 1 ≤ x 2 with p = 0.25. x 1 , 1 , x 1 , 2 , x 1 , 3 are all 0.25-stable for AC for the constraint, but x 1 , 4 is not, because its only AC-support, x 2 , 4 , has distance 0. The parameterized strategy p-LC [1] enforces, on each variable-value pair, either AC or some local consistency (LC) property strictly stronger than AC depending on the value of the parameter p. The idea is to enforce LC only on the variable-value pairs with few supports, approximated with the rank (< p) of the first found AC-support. We focus on the constraint-based version, pc-LC, where x i , v i is pc-LC if for every constraint c j ∈ cons(x i ), x i , v i is p-stable for AC on c j or x i , v i is LC on c j .…”

“…The adaptive level p(c j ) is defined by Balafrej et al [1] and recalled in Equation (1). The local consistency technique used here is the implementation of R( * ,2)C [12], apc-R( * ,2)C. Apply-R( * ,2)C (Algorithm 2) takes as input the list of tuples of a constraint on which R( * ,2)C must be enforced.…”

“…The above-mentioned approaches do not allow us to enforce different levels of consistency on the values in the domain of the same variable. To this end, Balafrej et al introduced adaptive parameterized consistency, which selects, for each value in the domain of a variable, one of two consistency levels based on the value of a parameter [1]. That parameter is determined by the rank of the support of the value in a constraint (assuming a fixed total ordering of the variables' domains), and updated depending on the weight of the constraint [5].…”

Visualizations to Summarize Search Behavior

“…However, its performance is unpredictable and can differ widely on similar instances. Further, maintaining a given consistency property during search has become a common practice [13,22,30,42,43] and new strategies for dynamically switching between consistency algorithms are being investigated [1][2][3]9,15,16,18,25,29,41,46,48]. While consistency algorithms can dramatically reduce the size of the search space, their impact on the CPU cost of search can vary widely.…”