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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



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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

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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

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.

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

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