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Characterization of Matrices with Bounded Graver Bases and Depth Parameters and Applications to Integer Programming

Authors Marcin Briański, Martin Koutecký, Daniel Král', Kristýna Pekárková, Felix Schröder

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Author Details

Marcin Briański
  • Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
Martin Koutecký
  • Computer Science Institute, Charles University, Prague, Czech Republic
Daniel Král'
  • Faculty of Informatics, Masaryk University, Brno, Czech Republic
Kristýna Pekárková
  • Faculty of Informatics, Masaryk University, Brno, Czech Republic
Felix Schröder
  • Institute of Mathematics, Technische Universität, Berlin, Germany

Funding

The third author was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 648509). This publication reflects only its authors' view; the ERC Executive Agency is not responsible for any use that may be made of the information it contains. The third and fourth authors were supported by the MUNI Award in Science and Humanities (MUNI/I/1677/2018) of the Grant Agency of Masaryk University. The second author was partially supported by Charles University project UNCE/SCI/004 and by the project by the project 19-27871X of GA ČR. First author was partially supported by the Polish National Science Center grant (BEETHOVEN; UMO-2018/31/G/ST1/03718).

Acknowledgements

All five authors would like to thank the Schloss Dagstuhl - Leibniz-Zentrum für Informatik for hospitality during the workshop "Sparsity in Algorithms, Combinatorics and Logic" in September 2021 where the work leading to the results contained in this paper was started.

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Marcin Briański, Martin Koutecký, Daniel Král', Kristýna Pekárková, and Felix Schröder. Characterization of Matrices with Bounded Graver Bases and Depth Parameters and Applications to Integer Programming. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 29:1-29:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ICALP.2022.29

Abstract

An intensive line of research on fixed parameter tractability of integer programming is focused on exploiting the relation between the sparsity of a constraint matrix A and the norm of the elements of its Graver basis. In particular, integer programming is fixed parameter tractable when parameterized by the primal tree-depth and the entry complexity of A, and when parameterized by the dual tree-depth and the entry complexity of A; both these parameterization imply that A is sparse, in particular, the number of its non-zero entries is linear in the number of columns or rows, respectively.
We study preconditioners transforming a given matrix to an equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the 𝓁₁-norm of the Graver basis is bounded by a function of the maximum 𝓁₁-norm of a circuit of A. We use our results to design a parameterized algorithm that constructs a matrix equivalent to an input matrix A that has small primal/dual tree-depth and entry complexity if such an equivalent matrix exists.
Our results yield parameterized algorithms for integer programming when parameterized by the 𝓁₁-norm of the Graver basis of the constraint matrix, when parameterized by the 𝓁₁-norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix equivalent to the constraint matrix.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial optimization
  • Mathematics of computing → Matroids and greedoids
  • Mathematics of computing → Combinatorial algorithms
Keywords
  • Integer programming
  • width parameters
  • matroids
  • Graver basis
  • tree-depth
  • fixed parameter tractability

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