Sparse solutions of sparse linear systems: Fixed-parameter tractability and an application of complex group testing
Artikel i vetenskaplig tidskrift, 2013

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 a 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 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.

group testing

problem kernel

parameterized algorithm

hitting set

sparse vector


linear system


Peter Damaschke

Chalmers, Data- och informationsteknik, Datavetenskap

Theoretical Computer Science

0304-3975 (ISSN)

Vol. 511 137-146



Diskret matematik


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