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

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14330%2F22%3A00126436" target="_blank" >RIV/00216224:14330/22:00126436 - isvavai.cz</a>
Alternative codes found
RIV/00216208:11320/22:10455466
Result on the web
<a href="https://epubs.siam.org/doi/10.1137/20M1353502" target="_blank" >https://epubs.siam.org/doi/10.1137/20M1353502</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1137/20M1353502" target="_blank" >10.1137/20M1353502</a>

Result language
angličtina
Original language name
Matrices of Optimal Tree-Depth and a Row-Invariant Parameterized Algorithm for Integer Programming
Original language description
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 dual tree-depth d and largest entry Δ are solvable in time g(d, Δ)poly(n) for some function g. However, the dual 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, and thus does not reflect its geometric structure. We prove that the minimum dual tree-depth of a row-equivalent matrix is equal to the branch-depth of the matroid defined by the columns of the matrix. We design a fixed parameter algorithm for computing branch-depth of matroids represented over a finite field and a fixed parameter algorithm for computing a row-equivalent matrix with minimum dual treedepth. Finally, we use these results to obtain an algorithm for integer programming running in time g(d∗, Δ)poly(n) where d∗ is the branch-depth of the constraint matrix; the branch-depth cannot be replaced by the more permissive notion of branch-width.
Czech name
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Czech description
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Type
J<sub>SC</sub> - Article in a specialist periodical, which is included in the SCOPUS database
CEP classification
—
OECD FORD branch
10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Project
<a href="/en/project/GX19-27871X" target="_blank" >GX19-27871X: Efficient approximation algorithms and circuit complexity</a><br>
Continuities
S - Specificky vyzkum na vysokych skolach

Publication year
2022
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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