Computing the spark: mixed-integer programming for the (vector) matroid girth problem

  • Andreas M. TillmannEmail author


We investigate the NP-hard problem of computing the spark of a matrix (i.e., the smallest number of linearly dependent columns), a key parameter in compressed sensing and sparse signal recovery. To that end, we identify polynomially solvable special cases, gather upper and lower bounding procedures, and propose several exact (mixed-)integer programming models and linear programming heuristics. In particular, we develop a branch and cut scheme to determine the girth of a matroid, focussing on the vector matroid case, for which the girth is precisely the spark of the representation matrix. Extensive numerical experiments demonstrate the effectiveness of our specialized algorithms compared to general-purpose black-box solvers applied to several mixed-integer programming models.


Matroid girth Spark Sparse recovery Compressed sensing Branch-and-cut Mixed-integer programming 

Mathematics Subject Classification

05B35 68Q17 90C10 90C11 90C57 94A15 



The author would like to thank Marc Pfetsch for inspiring discussions in the early stages of this research effort and his support in getting started working with the SCIP framework, the authors of [38, 75] for kindly providing code for their respective deterministic compressed sensing matrix construction routines, as well as the anonymous reviewer whose comments helped improve the paper.


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Authors and Affiliations

  1. 1.Visual Computing Institute and Chair of Operations ResearchRWTH Aachen UniversityAachenGermany

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