Effects of Dynamic Variable - Value Ordering Heuristics on the Search Space of Sudoku Modeled as a Constraint Satisfaction Problem

We carry out a detailed analysis of the effects of different dynamic variable and value ordering heuristics on the search space of Sudoku when the encoding method and the filtering algorithm are fixed. Our study starts by examining lexicographical variable and value ordering and evaluates different combinations of dynamic variable and value ordering heuristics. We eventually build up to a dynamic variable ordering heuristic that has two rounds of tie-breakers, where the second tie-breaker is a dynamic value ordering heuristic. We show that our method that uses this interlinked heuristic outperforms the previously studied ones with the same experimental setup. Overall, we conclude that constructing insightful dynamic variable ordering heuristics that also utilize a dynamic value ordering heuristic in their decision making process could drastically improve the search effort for some constraint satisfaction problems.

Exploring the Impact of Early Decisions in Variable Ordering for Constraint Satisfaction Problems

When solving constraint satisfaction problems (CSPs), it is a common practice to rely on heuristics to decide which variable should be instantiated at each stage of the search. But, this ordering influences the search cost. Even so, and to the best of our knowledge, no earlier work has dealt with how first variable orderings affect the overall cost. In this paper, we explore the cost of finding high-quality orderings of variables within constraint satisfaction problems. We also study differences among the orderings produced by some commonly used heuristics and the way bad first decisions affect the search cost. One of the most important findings of this work confirms the paramount importance of first decisions. Another one is the evidence that many of the existing variable ordering heuristics fail to appropriately select the first variable to instantiate. Another one is the evidence that many of the existing variable ordering heuristics fail to appropriately select the first variable to instantiate. We propose a simple method to improve early decisions of heuristics. By using it, performance of heuristics increases.

Experimental Matching of Instances to Heuristics for Constraint Satisfaction Problems

Constraint satisfaction problems are of special interest for the artificial intelligence and operations research community due to their many applications. Although heuristics involved in solving these problems have largely been studied in the past, little is known about the relation between instances and the respective performance of the heuristics used to solve them. This paper focuses on both the exploration of the instance space to identify relations between instances and good performing heuristics and how to use such relations to improve the search. Firstly, the document describes a methodology to explore the instance space of constraint satisfaction problems and evaluate the corresponding performance of six variable ordering heuristics for such instances in order to find regions on the instance space where some heuristics outperform the others. Analyzing such regions favors the understanding of how these heuristics work and contribute to their improvement. Secondly, we use the information gathered from the first stage to predict the most suitable heuristic to use according to the features of the instance currently being solved. This approach proved to be competitive when compared against the heuristics applied in isolation on both randomly generated and structured instances of constraint satisfaction problems.

BDD Ordering Heuristics for Classical Planning

Symbolic search using binary decision diagrams (BDDs) can often save large amounts of memory due to its concise representation of state sets. A decisive factor for this method's success is the chosen variable ordering. Generally speaking, it is plausible that dependent variables should be brought close together in order to reduce BDD sizes. In planning, variable dependencies are typically captured by means of causal graphs, and in preceding work these were taken as the basis for finding BDD variable orderings. Starting from the observation that the two concepts of "dependency" are actually quite different, we introduce a framework for assessing the strength of variable ordering heuristics in sub-classes of planning. It turns out that, even for extremely simple planning tasks, causal graph based variable orders may be exponentially worse than optimal. Experimental results on a wide range of variable ordering variants corroborate our theoretical findings. Furthermore, we show that dynamic reordering is much more effective at reducing BDD size, but it is not cost-effective due to a prohibitive runtime overhead. We exhibit the potential of middle-ground techniques, running dynamic reordering until simple stopping criteria hold.