A Global Constraint for the Exact Cover Problem: Application to Conceptual Clustering

We introduce the exactCover global constraint dedicated to the exact cover problem, the goal of which is to select subsets such that each element of a given set belongs to exactly one selected subset. This NP-complete problem occurs in many applications, and we more particularly focus on a conceptual clustering application. We introduce three propagation algorithms for exactCover, called Basic, DL, and DL+: Basic ensures the same level of consistency as arc consistency on a classical decomposition of exactCover into binary constraints, without using any specific data structure; DL ensures the same level of consistency as Basic but uses Dancing Links to efficiently maintain the relation between elements and subsets; and DL+ is a stronger propagator which exploits an extra property to filter more values than DL. We also consider the case where the number of selected subsets is constrained to be equal to a given integer variable k, and we show that this may be achieved either by combining exactCover with existing constraints, or by designing a specific propagator that integrates algorithms designed for the NValues constraint. These different propagators are experimentally evaluated on conceptual clustering problems, and they are compared with state-of-the-art declarative approaches. In particular, we show that our global constraint is competitive with recent ILP and CP models for mono-criterion problems, and it has better scale-up properties for multi-criteria problems.

Descriptive Clustering: ILP and CP Formulations with Applications

In many settings just finding a good clustering is insufficient and an explanation of the clustering is required. If the features used to perform the clustering are interpretable then methods such as conceptual clustering can be used. However, in many applications this is not the case particularly for image, graph and other complex data. Here we explore the setting where a set of interpretable discrete tags for each instance is available. We formulate the descriptive clustering problem as a bi-objective optimization to simultaneously find compact clusters using the features and to describe them using the tags. We present our formulation in a declarative platform and show it can be integrated into a standard iterative algorithm to find all Pareto optimal solutions to the two objectives. Preliminary results demonstrate the utility of our approach on real data sets for images and electronic health care records and that it outperforms single objective and multi-view clustering baselines.