Matrices of Optimal Tree-Depth and Row-Invariant Parameterized Algorithm for Integer Programming
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.
Matroid algorithms
width parameters
integer programming
fixed parameter tractability
branch-width
branch-depth
Mathematics of computing~Combinatorial optimization
Mathematics of computing~Matroids and greedoids
Mathematics of computing~Combinatorial algorithms
26:1-26:19
Track A: Algorithms, Complexity and Games
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.
A full version is available at https://arxiv.org/abs/1907.06688.
Timothy F. N.
Chan
Timothy F. N. Chan
School of Mathematical Sciences, Monash University, Melbourne, Australia
Mathematics Institute and DIMAP, University of Warwick, Coventry, UK
Jacob W.
Cooper
Jacob W. Cooper
Faculty of Informatics, Masaryk University, Brno, Czech Republic
Martin
Koutecký
Martin Koutecký
Computer Science Institute, Charles University, Prague, Czech Republic
Daniel
Král'
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
Kristýna Pekárková
Faculty of Informatics, Masaryk University, Brno, Czech Republic
10.4230/LIPIcs.ICALP.2020.26
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Timothy F. N. Chan, Jacob W. Cooper, Martin Koutecký, Daniel Král', and Kristýna Pekárková
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