Sparse solutions of sparse linear systems: Fixed-parameter tractability and an application of complex group testing

Abstract

A vector with at most k nonzeros is called k-sparse. We show that enumerating the support vectors of k-sparse solutions to a system Ax=b of r-sparse linear equations (i.e., where the rows of A are r-sparse) is fixed-parameter tractable (FPT) in the combined parameter r,k. We give different branching algorithms based on the close relationship to the hitting set problem in fixed-rank hypergraphs. For r=2 the problem is simple. For 0,1-matrices A we can also compute an O(rkr) kernel. For systems of linear inequalities we get an FPT result in the combined parameter d,k, where d is the total number of minimal solutions. This is achieved by interpreting the problem as a case of group testing in the complex model. The problems stem from the reconstruction of chemical mixtures by observable reaction products.