Principles and Practice of Constraint Programming

TLDR: This work proposes a global ranking constraint and shows that GAC can be achieved in polynomial time and proposes an Oðn3 log nÞ algorithm for achieving RC as well as an efficient quadratic algorithm offering a better tradeoff.

Abstract: In many problems we want to reason about the ranking of items. For example, in information retrieval, when aggregating several search results, we may have ties and consequently rank orders. (e.g. [2, 3]). As a second example, we may wish to construct an overall ranking of tennis player based on pairwise comparisons between players. One principled method for constructing a ranking is the Kemeny distance [5] as this is the unique scheme that is neutral, consistent, and Condorcet. Unfortunately, determining this ranking is NP-hard, and remains so when we permit ties in the input or output [4]. As a third example, tasks in a scheduling problem may run in parallel, resulting in a ranking. In a ranking, unlike a permutation, we can have ties. Thus, 12225 is a ranking whilst 12345 is a permutation. To reason about permutations, we have efficient and effective global constraints. Regin [7] proposed an Oðn4Þ GAC propagator for permutations. For BC, there is an even faster Oðn log nÞ propagator [6]. Every constraint toolkit now provides propagators for permutation constraints. Surprisingly, ranking constraints are not yet supported. In [1], we tackle this weakness by proposing a global ranking constraint. We show that simple decompositions of this constraint hurt pruning. We then show that GAC can be achieved in polynomial time and we propose an Oðn3 log nÞ algorithm for achieving RC as well as an efficient quadratic algorithm offering a better tradeoff.