Domain-Heuristics for Arc-Consistency Algorithms

Arc-consistency algorithms are widely used to prune the search-space of Constraint Satisfaction Problems (CSPs). They use support-checks (also known as consistency-checks) to find out about the properties of CSPs. They use arc-heuristics to select the next constraint and domain-heuristics to select the next values for their next support-check. We will investigate the effects of domain- heuristics by studying the average time-complexity of two arc-consistency algorithms which use different domain-heuristics. We will assume that there are only two variables. The first algorithm, called \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{L} $$\end{document} , uses a lexicographical heuristic. The second algorithm, called \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{D} $$\end{document}, uses a heuristic based on the notion of a double- support check.We will discuss the consequences of our simplification about the number of variables in the CSP and we will carry out a case-study for the case where the domain-sizes of the variables is two.We will present relatively simple formulae for the exact average time-complexity of both algorithms as well as simple bounds. As a and b become large \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{L} $$\end{document} will require about 2a + 2b − 2 log2(a) − 0.665492 checks on average, where a and b are the domain-sizes and log2(·) is the base- 2 logarithm. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{D} $$\end{document} requires an average number of support-checks which is below 2 max(a, b) + 2 if a + b ≥ 14. Our results demonstrate that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \