Matrices of optimal tree-depth and a row-invariant parameterized algorithm for integer programming

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14330%2F20%3A00116613" target="_blank" >RIV/00216224:14330/20:00116613 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.4230/LIPIcs.ICALP.2020.26" target="_blank" >http://dx.doi.org/10.4230/LIPIcs.ICALP.2020.26</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4230/LIPIcs.ICALP.2020.26" target="_blank" >10.4230/LIPIcs.ICALP.2020.26</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Matrices of optimal tree-depth and a row-invariant parameterized algorithm for integer programming

Popis výsledku v původním jazyce

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 D are solvable in time g(d,D)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 tree-depth. Finally, we use these results to obtain an algorithm for integer programming running in time g(d*,D)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.

Název v anglickém jazyce

Matrices of optimal tree-depth and a row-invariant parameterized algorithm for integer programming

Popis výsledku anglicky

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 D are solvable in time g(d,D)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 tree-depth. Finally, we use these results to obtain an algorithm for integer programming running in time g(d*,D)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.

Druh

D - Stať ve sborníku

CEP obor

—

OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2020

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název statě ve sborníku

Proceedings of the 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), p. "26:1"-"26:19", 19 pp. 2020.

ISBN

9783959771382

ISSN

1868-8969

e-ISSN

—

Počet stran výsledku

19

Strana od-do

„26:1“-„26:19“

Název nakladatele

Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik

Místo vydání

Dagstuhl, Germany

Místo konání akce

Dagstuhl, Germany

Datum konání akce

1. 1. 2020

Typ akce podle státní příslušnosti

WRD - Celosvětová akce

Kód UT WoS článku

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