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

J 2022

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

CHAN, Timothy Fong Nam, Jacob COOPER, Martin KOUTECKÝ, Daniel KRÁĽ, Kristýna PEKÁRKOVÁ et. al.

Základní údaje

Originální název

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

Autoři

CHAN, Timothy Fong Nam (826 Velká Británie a Severní Irsko), Jacob COOPER (826 Velká Británie a Severní Irsko, domácí), Martin KOUTECKÝ (203 Česká republika), Daniel KRÁĽ (203 Česká republika, garant, domácí) a Kristýna PEKÁRKOVÁ (203 Česká republika, domácí)

Vydání

SIAM JOURNAL ON COMPUTING, UNITED STATES, SIAM PUBLICATIONS, 2022, 0097-5397

Další údaje

Jazyk

angličtina

Typ výsledku

Článek v odborném periodiku

Obor

10201 Computer sciences, information science, bioinformatics

Stát vydavatele

Spojené státy

Utajení

není předmětem státního či obchodního tajemství

Odkazy

URL

Impakt faktor

Impact factor: 1.600

Kód RIV

RIV/00216224:14330/22:00126436

Organizační jednotka

Fakulta informatiky

DOI

http://dx.doi.org/10.1137/20M1353502

UT WoS

001130401900013

Klíčová slova anglicky

branch-depth; fixed-parameter tractability; integer programming; matroids; tree-depth

Příznaky

Mezinárodní význam, Recenzováno

Změněno: 13. 5. 2024 16:44, RNDr. Pavel Šmerk, Ph.D.

Anotace

V originále

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.

Návaznosti

MUNI/A/1145/2021, interní kód MU

Název: Rozsáhlé výpočetní systémy: modely, aplikace a verifikace XI. (Akronym: SV-FI MAV XI.)

Investor: Masarykova univerzita, Rozsáhlé výpočetní systémy: modely, aplikace a verifikace XI.

MUNI/I/1677/2018, interní kód MU

Název: MUNI AWARD in Science and Humanitites 1 (Akronym: MASH 1)

Investor: Masarykova univerzita, MUNI AWARD in Science and Humanitites 1, MASH - MUNI Award in Science and Humanities

Citovat

CHAN, Timothy Fong Nam, Jacob COOPER, Martin KOUTECKÝ, Daniel KRÁĽ a Kristýna PEKÁRKOVÁ. Matrices of Optimal Tree-Depth and a Row-Invariant Parameterized Algorithm for Integer Programming. SIAM JOURNAL ON COMPUTING. UNITED STATES: SIAM PUBLICATIONS, 2022, roč. 51, č. 3, s. 664-700. ISSN 0097-5397. Dostupné z: https://dx.doi.org/10.1137/20M1353502.

@article{2211686,
   author = {Chan, Timothy Fong Nam and Cooper, Jacob and Koutecký, Martin and Kráľ, Daniel and Pekárková, Kristýna},
   article_location = {UNITED STATES},
   article_number = {3},
   doi = {http://dx.doi.org/10.1137/20M1353502},
   keywords = {branch-depth; fixed-parameter tractability; integer programming; matroids; tree-depth},
   language = {eng},
   issn = {0097-5397},
   journal = {SIAM JOURNAL ON COMPUTING},
   title = {Matrices of Optimal Tree-Depth and a Row-Invariant Parameterized Algorithm for Integer Programming},
   url = {https://epubs.siam.org/doi/10.1137/20M1353502},
   volume = {51},
   year = {2022}
}

TY  - JOUR
ID  - 2211686
AU  - Chan, Timothy Fong Nam - Cooper, Jacob - Koutecký, Martin - Kráľ, Daniel - Pekárková, Kristýna
PY  - 2022
TI  - Matrices of Optimal Tree-Depth and a Row-Invariant Parameterized Algorithm for Integer Programming
JF  - SIAM JOURNAL ON COMPUTING
VL  - 51
IS  - 3
SP  - 664-700
EP  - 664-700
PB  - SIAM PUBLICATIONS
SN  - 00975397
KW  - branch-depth
KW  - fixed-parameter tractability
KW  - integer programming
KW  - matroids
KW  - tree-depth
UR  - https://epubs.siam.org/doi/10.1137/20M1353502
N2  - 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.
ER  -

CHAN, Timothy Fong Nam, Jacob COOPER, Martin KOUTECKÝ, Daniel KRÁĽ a Kristýna PEKÁRKOVÁ. Matrices of Optimal Tree-Depth and a Row-Invariant Parameterized Algorithm for Integer Programming. \textit{SIAM JOURNAL ON COMPUTING}. UNITED STATES: SIAM PUBLICATIONS, 2022, roč.~51, č.~3, s.~664-700. ISSN~0097-5397. Dostupné z: https://dx.doi.org/10.1137/20M1353502.

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