eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-06-29
26:1
26:19
10.4230/LIPIcs.ICALP.2020.26
article
Matrices of Optimal Tree-Depth and Row-Invariant Parameterized Algorithm for Integer Programming
Chan, Timothy F. N.
1
2
Cooper, Jacob W.
3
Koutecký, Martin
4
Král', Daniel
3
5
Pekárková, Kristýna
3
School of Mathematical Sciences, Monash University, Melbourne, Australia
Mathematics Institute and DIMAP, University of Warwick, Coventry, UK
Faculty of Informatics, Masaryk University, Brno, Czech Republic
Computer Science Institute, Charles University, Prague, Czech Republic
Mathematics Institute, DIMAP and Department of Computer Science, University of Warwick, Coventry, UK
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol168-icalp2020/LIPIcs.ICALP.2020.26/LIPIcs.ICALP.2020.26.pdf
Matroid algorithms
width parameters
integer programming
fixed parameter tractability
branch-width
branch-depth