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%2F22%3A00126436" target="_blank" >RIV/00216224:14330/22:00126436 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/00216208:11320/22:10455466

Výsledek na webu

<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>

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 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.

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 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.

Druh

J_SC - Článek v periodiku v databázi SCOPUS

CEP obor

—

OECD FORD obor

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

Projekt

<a href="/cs/project/GX19-27871X" target="_blank" >GX19-27871X: Efektivní aproximační algoritmy a obvodová složitost</a><br>

Návaznosti

S - Specificky vyzkum na vysokych skolach

Rok uplatnění

2022

Kód důvěrnosti údajů

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

Název periodika

SIAM JOURNAL ON COMPUTING

ISSN

0097-5397

e-ISSN

1095-7111

Svazek periodika

51

Číslo periodika v rámci svazku

3

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

37

Strana od-do

664-700

Kód UT WoS článku

—

EID výsledku v databázi Scopus

2-s2.0-85132215614