Robert J. Woodward

Revisiting Neighborhood Inverse Consistency on Binary CSPs

Our goal is to investigate the definition and application of strong consistency properties on the dual graphs of binary Constraint Satisfaction Problems (CSPs). As a first step in that direction, we study the structure of the dual graph of binary CSPs, and show how it can be arranged in a triangle-shaped grid. We then study, in this context, Relational Neighborhood Inverse Consistency (RNIC), which is a consistency property that we had introduced for non-binary CSPs [17]. We discuss how the structure of the dual graph of binary CSPs affects the consistency level enforced by RNIC. Then, we compare, both theoretically and empirically, RNIC to Neighborhood Inverse Consistency (NIC) and strong Conservative Dual Consistency (sCDC), which are higher-level consistency properties useful for solving difficult problem instances. We show that all three properties are pairwise incomparable.

Improving Relational Consistency Algorithms Using Dynamic Relation Partitioning

Relational consistency algorithms are instrumental for solving difficult instances of Constraint Satisfaction Problems (CSPs), often allowing backtrack-free search. In this paper, we improve an algorithm for enforcing relational consistency by exploiting the property that the constraints of the dual encoding of a CSP are piecewise functional. This property allows us to partition a CSP relation into blocks of equivalent tuples at varying levels of granularity. Our new algorithm dynamically exploits those partitions. Our experiments show a significant improvement of the processing time for enforcing relational consistency.

Adaptive Parameterized Consistency for Non-binary CSPs by Counting Supports

Determining the appropriate level of local consistency to enforce on a given instance of a Constraint Satisfaction Problem (CSP) is not an easy task. However, selecting the right level may determine our ability to solve the problem. Adaptive parameterized consistency was recently proposed for binary CSPs as a strategy to dynamically select one of two local consistencies (i.e., AC and maxRPC). In this paper, we propose a similar strategy for non-binary table constraints to select between enforcing GAC and pairwise consistency. While the former strategy approximates the supports by their rank and requires that the variables domains be ordered, our technique removes those limitations. We empirically evaluate our approach on benchmark problems to establish its advantages.

Relational consistency by constraint filtering

In this paper, we propose a new algorithm for enforcing relational consistency on every set of k constraints of a finite Constraint Satisfaction Problem (CSP). This algorithm operates by filtering the constraint while leaving the topology of the graph unchanged. We study the resulting relational consistency property and compare it to existing ones. We evaluate the effectiveness of our algorithm in a search procedure for solving CSPs and demonstrate the applicability, effectiveness, and usefulness of enforcing high levels of consistency.

Solving Sudoku with Consistency: A Visual and Interactive Approach

We describe an online, interactive system with a graphical interface to illustrate the power and operation of consistency algorithms in a friendly and popular context, namely, solving Sudoku puzzles. Our tool implements algorithms for enforcing five (domain-based) consistency properties on binary and non-binary constraint models. Our tool is useful for research, education, and outreach. From a scientific standpoint, we propose a new consistency property that can solve the hardest known 9×9 Sudoku instances without search, but leave open the question of the lowest level of consistency needed to solve every 9×9 Sudoku puzzle. We have used the current tool and its predecessor in the classroom to introduce students to modeling problems with constraints, explain consistency properties, and illustrate the operations of constraint propagation and lookahead. Finally, we have also used this tool during outreach activities to demystify AI to children and the general public and show them how computers think.

A Reactive Strategy for High-Level Consistency During Search

Constraint propagation during backtrack search significantly improves the performance of solving a Constraint Satisfaction Problem. While Generalized Arc Consistency (GAC) is the most popular level of propagation, higher-level consistencies (HLC) are needed to solve difficult instances. Deciding to enforce an HLC instead of GAC remains the topic of active research. We propose a simple and effective strategy that reactively triggers an HLC by monitoring search performance: When search starts thrashing, we trigger an HLC, then conservatively revert to GAC. We detect thrashing by counting the number of backtracks at each level of the search tree and geometrically adjust the frequency of triggering an HLC based on its filtering effectiveness. We validate our approach on benchmark problems using Partition-One Arc-Consistency as an HLC. However, our strategy is generic and can be used with other higher-level consistency algorithms.