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

Peter Damaschke
2013 Theoretical Computer Science  
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(rk r ) kernel. For
more » ... stems 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 interpeting 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.
doi:10.1016/j.tcs.2012.07.001 fatcat:pht3ay5bnve5zedkq34h5utjue