A Reactive Strategy for High-Level Consistency During Search

Abstract

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.

Visualizations to Summarize Search Behavior

“…Both POAC and (strong) PPC are strictly stronger than GAC. In order to control their cost, we enforce the HLCs in a 'selective/adaptive' manner using the PrePeak + strategy of Woodward et al [46] for POAC (denoted PrePeak + -POAC) and the Directional Partial Path Consistency of Woodward [45] (denoted DPPC + ). PrePeak + triggers POAC by watching the numbers of backtracks and follows geometric laws to increase or decrease the calls to POAC.…”

“…Tracing the search effort by depth of the search tree was first proposed for the number of constraint checks and values removed per search level (Epstein et al [16]) and for the number of nodes visited (Simonis et al [39] in CP-Viz, also used for solving a packing problem [40]). More recently, Woodward et al [46] propose to focus on the number of backtracks per depth (BpD) to assess the severity of thrashing and on the number of calls to an HLC per depth (CpD) to explain the cost of enforcing an HLC, see Fig. 1.…”

“…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.…”

“…Building on the visualizations proposed by Woodward et al [46] for describing search performance, we introduce the following contributions: 1. A criterion for computing the distance between two samples, quantifying the change in search behavior.…”

“…From Woodward et al[46]: Number of backtracks (BpD) and calls (CpD) to POAC per depth at the end of search for benchmark instance pseudo-aim-200-1-6-4 using dom/wdeg. Left: GAC.…”