Characterization of Matrices with Bounded Graver Bases and Depth Parameters and Applications to Integer Programming

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F22%3A10455461" target="_blank" >RIV/00216208:11320/22:10455461 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/00216224:14330/22:00126328

Výsledek na webu

<a href="https://doi.org/10.4230/LIPIcs.ICALP.2022.29" target="_blank" >https://doi.org/10.4230/LIPIcs.ICALP.2022.29</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4230/LIPIcs.ICALP.2022.29" target="_blank" >10.4230/LIPIcs.ICALP.2022.29</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Characterization of Matrices with Bounded Graver Bases and Depth Parameters and Applications to Integer Programming

Popis výsledku v původním jazyce

An intensive line of research on fixed parameter tractability of integer programming is focused on exploiting the relation between the sparsity of a constraint matrix A and the norm of the elements of its Graver basis. In particular, integer programming is fixed parameter tractable when parameterized by the primal tree-depth and the entry complexity of A, and when parameterized by the dual tree-depth and the entry complexity of A; both these parameterization imply that A is sparse, in particular, the number of its non-zero entries is linear in the number of columns or rows, respectively. We study preconditioners transforming a given matrix to an equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the ℓ1-norm of the Graver basis is bounded by a function of the maximum ℓ1-norm of a circuit of A. We use our results to design a parameterized algorithm that constructs a matrix equivalent to an input matrix A that has small primal/dual tree-depth and entry complexity if such an equivalent matrix exists. Our results yield parameterized algorithms for integer programming when parameterized by the ℓ1-norm of the Graver basis of the constraint matrix, when parameterized by the ℓ1-norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix equivalent to the constraint matrix.

Název v anglickém jazyce

Characterization of Matrices with Bounded Graver Bases and Depth Parameters and Applications to Integer Programming

Popis výsledku anglicky

An intensive line of research on fixed parameter tractability of integer programming is focused on exploiting the relation between the sparsity of a constraint matrix A and the norm of the elements of its Graver basis. In particular, integer programming is fixed parameter tractable when parameterized by the primal tree-depth and the entry complexity of A, and when parameterized by the dual tree-depth and the entry complexity of A; both these parameterization imply that A is sparse, in particular, the number of its non-zero entries is linear in the number of columns or rows, respectively. We study preconditioners transforming a given matrix to an equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the ℓ1-norm of the Graver basis is bounded by a function of the maximum ℓ1-norm of a circuit of A. We use our results to design a parameterized algorithm that constructs a matrix equivalent to an input matrix A that has small primal/dual tree-depth and entry complexity if such an equivalent matrix exists. Our results yield parameterized algorithms for integer programming when parameterized by the ℓ1-norm of the Graver basis of the constraint matrix, when parameterized by the ℓ1-norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix equivalent to the constraint matrix.

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

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

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2022

Kód důvěrnosti údajů

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

Název statě ve sborníku

Leibniz International Proceedings in Informatics, LIPIcs

ISBN

978-3-95977-235-8

ISSN

—

e-ISSN

—

Počet stran výsledku

20

Strana od-do

1-20

Název nakladatele

Schloss Dagstuhl

Místo vydání

Dagstuhl

Místo konání akce

Paříž

Datum konání akce

4. 7. 2022

Typ akce podle státní příslušnosti

WRD - Celosvětová akce

Kód UT WoS článku

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