Characterization of matrices with bounded Graver bases and depth parameters and applications to integer programming

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F24%3A10490811" target="_blank" >RIV/00216208:11320/24:10490811 - isvavai.cz</a>

Alternative codes found

RIV/00216224:14330/24:00137365

Result on the web

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=cL5ThN_qTy" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=cL5ThN_qTy</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s10107-023-02048-x" target="_blank" >10.1007/s10107-023-02048-x</a>

Result language

angličtina

Original language name

Characterization of matrices with bounded Graver bases and depth parameters and applications to integer programming

Original language description

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 a row-equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse row-equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the l(1)-norm of the Graver basis is bounded by a function of the maximum l(1)-norm of a circuit of A. We use our results to design a parameterized algorithm that constructs a matrix row-equivalent to an input matrix A that has small primal/dual tree-depth and entry complexity if such a row-equivalent matrix exists. Our results yield parameterized algorithms for integer programming when parameterized by the l(1)-norm of the Graver basis of the constraint matrix, when parameterized by the l(1)-norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix row-equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix row-equivalent to the constraint matrix.

Czech name

—

Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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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/GX23-04949X" target="_blank" >GX23-04949X: Fundamental questions of discrete geometry</a><br>

Continuities

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

Publication year

2024

Confidentiality

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

Name of the periodical

Mathematical Programming, Series A

ISSN

0025-5610

e-ISSN

1436-4646

Volume of the periodical

208

Issue of the periodical within the volume

1-2

Country of publishing house

DE - GERMANY

Number of pages

35

Pages from-to

497-531

UT code for WoS article

001145544400002

EID of the result in the Scopus database

2-s2.0-85182671269