Mining a Maximum Weighted Set of Disjoint Submatrices

Abstract

The objective of the maximum weighted set of disjoint submatrices problem is to discover K disjoint submatrices that together cover the largest sum of entries of an input matrix. It has many practical data-mining applications, as the related biclustering problem, such as gene module discovery in bioinformatics. It differs from the maximum-weighted submatrix coverage problem introduced in [6] by the explicit formulation of disjunction constraints: submatrices must not overlap. In other words, all matrix entries must be covered by at most one submatrix. The particular case of $K=1$, called the maximal-sum submatrix problem, was successfully tackled with constraint programming in [5]. Unfortunately, the case of $K > 1$ is more challenging to solve as the selection of rows cannot be decided in polynomial time solely from the selection of K sets of columns. It can be proved to be $\mathcal {NP}$-hard. We introduce a hybrid column generation approach using constraint programming to generate columns. It is compared to a standard mixed integer linear programming (MILP) through experiments on synthetic datasets. Overall, fast and valuable solutions are found by column generation while the MILP approach cannot handle a large number of variables and constraints.