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

Peter Damaschke (Institutionen för data- och informationsteknik, Datavetenskap, Algoritmer (Chalmers) ; Institutionen för data- och informationsteknik, Datavetenskap, Bioinformatik (Chalmers))
Theoretical Computer Science (0304-3975). Vol. 511 (2013), p. 137-146.
[Artikel, refereegranskad vetenskaplig]

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.

Nyckelord: sparse vector, linear system, hitting set, parameterized algorithm, enumeration, problem kernel, group testing

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Denna post skapades 2013-11-29. Senast ändrad 2015-02-11.
CPL Pubid: 187871


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