Document Open Access Logo

Matrices of Optimal Tree-Depth and Row-Invariant Parameterized Algorithm for Integer Programming

Authors Timothy F. N. Chan, Jacob W. Cooper, Martin Koutecký, Daniel Král', Kristýna Pekárková

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2020.26.pdf
  • Filesize: 0.52 MB
  • 19 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial optimization
  • Mathematics of computing → Matroids and greedoids
  • Mathematics of computing → Combinatorial algorithms
Keywords
  • Matroid algorithms
  • width parameters
  • integer programming
  • fixed parameter tractability
  • branch-width
  • branch-depth

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

A long line of research on fixed parameter tractability of integer programming culminated with showing that integer programs with n variables and a constraint matrix with tree-depth d and largest entry Δ are solvable in time g(d,Δ) poly(n) for some function g, i.e., fixed parameter tractable when parameterized by tree-depth d and Δ. However, the tree-depth of a constraint matrix depends on the positions of its non-zero entries and thus does not reflect its geometric structure. In particular, tree-depth of a constraint matrix is not preserved by row operations, i.e., a given integer program can be equivalent to another with a smaller dual tree-depth.
We prove that the branch-depth of the matroid defined by the columns of the constraint matrix is equal to the minimum tree-depth of a row-equivalent matrix. We also design a fixed parameter algorithm parameterized by an integer d and the entry complexity of an input matrix that either outputs a matrix with the smallest dual tree-depth that is row-equivalent to the input matrix or outputs that there is no matrix with dual tree-depth at most d that is row-equivalent to the input matrix. Finally, we use these results to obtain a fixed parameter algorithm for integer programming parameterized by the branch-depth of the input constraint matrix and the entry complexity. The parameterization by branch-depth cannot be replaced by the more permissive notion of branch-width.

Cite As Get BibTex

Timothy F. N. Chan, Jacob W. Cooper, Martin Koutecký, Daniel Král', and Kristýna Pekárková. Matrices of Optimal Tree-Depth and Row-Invariant Parameterized Algorithm for Integer Programming. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 26:1-26:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.26

Author Details

Timothy F. N. Chan
  • School of Mathematical Sciences, Monash University, Melbourne, Australia
  • Mathematics Institute and DIMAP, University of Warwick, Coventry, UK
Jacob W. Cooper
  • Faculty of Informatics, Masaryk University, Brno, Czech Republic
Martin Koutecký
  • Computer Science Institute, Charles University, Prague, Czech Republic
Daniel Král'
  • Faculty of Informatics, Masaryk University, Brno, Czech Republic
  • Mathematics Institute, DIMAP and Department of Computer Science, University of Warwick, Coventry, UK
Kristýna Pekárková
  • Faculty of Informatics, Masaryk University, Brno, Czech Republic

Funding

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

References

  1. Matthias Aschenbrenner and Raymond Hemmecke. Finiteness theorems in stochastic integer programming. Foundations of Computational Mathematics, 7(2):183-227, 2007. Google Scholar
  2. Lin Chen and Dániel Marx. Covering a tree with rooted subtrees-parameterized and approximation algorithms. In 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 2801-2820. SIAM, 2018. Google Scholar
  3. William H. Cunningham and Jim Geelen. On integer programming and the branch-width of the constraint matrix. In Integer Programming and Combinatorial Optimization, 12th International IPCO Conference, pages 158-166, 2007. Google Scholar
  4. Matt DeVos, O-joung Kwon, and Sang-il Oum. Branch-depth: Generalizing tree-depth of graphs. preprint, 2019. URL: http://arxiv.org/abs/1903.11988.
  5. Pavel Dvořák, Eduard Eiben, Robert Ganian, Dušan Knop, and Sebastian Ordyniak. Solving integer linear programs with a small number of global variables and constraints. In 26th International Joint Conference on Artificial Intelligence, pages 607-613. AAAI Press, 2017. Google Scholar
  6. Friedrich Eisenbrand, Christoph Hunkenschröder, and Kim-Manuel Klein. Faster Algorithms for Integer Programs with Block Structure. In 45th International Colloquium on Automata, Languages, and Programming, ICALP, pages 49:1-49:13, 2018. Google Scholar
  7. Friedrich Eisenbrand, Christoph Hunkenschröder, Kim-Manuel Klein, Martin Koutecký, Asaf Levin, and Shmuel Onn. An algorithmic theory of integer programming. preprint, 2019. URL: http://arxiv.org/abs/1904.01361.
  8. Fedor V. Fomin, Fahad Panolan, M. S. Ramanujan, and Saket Saurabh. On the optimality of pseudo-polynomial algorithms for integer programming. In 26th Annual European Symposium on Algorithms, ESA, pages 31:1-31:13, 2018. Google Scholar
  9. Robert Ganian and Sebastian Ordyniak. The complexity landscape of decompositional parameters for ILP. Artificial Intelligence, 2018. Google Scholar
  10. Robert Ganian, Sebastian Ordyniak, and M. S. Ramanujan. Going beyond primal treewidth for (M)ILP. In 31st AAAI Conference on Artificial Intelligence, pages 815-821, 2017. Google Scholar
  11. Tomas Gavenčiak, Daniel Král', and Sang-il Oum. Deciding first order properties of matroids. In 39th International Colloquium Automata, Languages, and Programming, ICALP, volume 7392 of Lecture Notes in Computer Science, pages 239-250, 2012. Google Scholar
  12. Jim Geelen, Albertus Gerards, Neil Robertson, and Geoff Whittle. On the excluded minors for the matroids of branch-width k. Journal of Combinatorial Theory, Series B, 88:261-265, 2003. Google Scholar
  13. Paul Halmos. Finite-Dimensional Vector Spaces. Undergraduate Texts in Mathematics. Springer, 1993. Google Scholar
  14. Raymond Hemmecke, Matthias Köppe, and Robert Weismantel. Graver basis and proximity techniques for block-structured separable convex integer minimization problems. Mathematical Programming, 145:1-18, 2014. Google Scholar
  15. Raymond Hemmecke, Shmuel Onn, and Lyubov Romanchuk. N-fold integer programming in cubic time. Mathematical Programming, 137:325-341, 2013. Google Scholar
  16. Petr Hliněný. Branch-width, parse trees, and monadic second-order logic for matroids. In Helmut Alt and Michel Habib, editors, 20th Annual Symposium on Theoretical Aspects of Computer Science, STACS, volume 2607 of LNCS, pages 319-330, 2003. Google Scholar
  17. Petr Hliněný. On matroid properties definable in the MSO logic. In Branislav Rovan and Peter Vojtáš, editors, 27th International Symposium on Mathematical Foundations of Computer Science, MFCS, volume 2747 of LNCS, pages 470-479, 2003. Google Scholar
  18. Petr Hliněný. Branch-width, parse trees, and monadic second-order logic for matroids. Journal of Combinatorial Theory, Series B, 96(3):325-351, 2006. Google Scholar
  19. Petr Hliněný and Sang-il Oum. Finding branch-decompositions and rank-decompositions. In Lars Arge, Michael Hoffmann, and Emo Welzl, editors, 15th Annual European Symposium, ESA, volume 4698 of LNCS, pages 163-174, 2007. Google Scholar
  20. Petr Hliněný and Sang-il Oum. Finding branch-decompositions and rank-decompositions. SIAM Journal on Computing, 38(3):1012-1032, 2008. Google Scholar
  21. Ravi Kannan. Minkowski’s convex body theorem and integer programming. Mathematics of Operations Research, 12(3):415–440, August 1987. Google Scholar
  22. František Kardoš, Daniel Král', Anita Liebenau, and Lukáš Mach. First order convergence of matroids. preprint, 2015. URL: http://arxiv.org/abs/1501.06518.
  23. František Kardoš, Daniel Král', Anita Liebenau, and Lukáš Mach. First order convergence of matroids. Eur. J. Comb, 59:150-168, 2017. Google Scholar
  24. Dušan Knop, Martin Koutecký, and Matthias Mnich. Voting and bribing in single-exponential time. In 34th Symposium on Theoretical Aspects of Computer Science, STACS, volume 66, pages 46:1-46:14, 2017. Google Scholar
  25. Martin Koutecký, Asaf Levin, and Shmuel Onn. A parameterized strongly polynomial algorithm for block structured integer programs. In 45th International Colloquium on Automata, Languages, and Programming, ICALP, volume 107, pages 85:1-85:14, 2018. Google Scholar
  26. Hendrik W. Lenstra, Jr. Integer programming with a fixed number of variables. Mathematics of Operations Research, 8(4):538-548, 1983. Google Scholar
  27. Susan Margulies, Jing Ma, and Illya V. Hicks. The Cunningham-Geelen method in practice: Branch-decompositions and integer programming. INFORMS Journal on Computing, 25(4):599-610, 2013. Google Scholar
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact