Computational complexity and ilp models for pattern problems in the logical analysis of data

Logical Analysis of Data is a procedure aimed at identifying relevant features in data sets with both positive and negative samples. The goal is to build Boolean formulas, represented by strings over {0,1,-} called patterns, which can be used to classify new samples as positive or negative. Since a data set can be explained in alternative ways, many computational problems arise related to the choice of a particular set of patterns. In this paper we study the computational complexity of several of these pattern problems (showing that they are, in general, computationally hard) and we propose some integer programming models that appear to be effective. We describe an ILP model for finding the minimum-size set of patterns explaining a given set of samples and another one for the problem of determining whether two sets of patterns are equivalent, i.e., they explain exactly the same samples. We base our first model on a polynomial procedure that computes all patterns compatible with a given set of samples. Computational experiments substantiate the effectiveness of our models on fairly large instances. Finally, we conjecture that the existence of an effective ILP model for finding a minimum-size set of patterns equivalent to a given set of patterns is unlikely, due to the problem being NP-hard and co-NP-hard at the same time.