2026-07-22T14:20:40Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:16370 2024-03-06T10:57:18Z ddc:004 open_access

Characterization of Matrices with Bounded Graver Bases and Depth Parameters and Applications to Integer Programming Briański, Marcin Koutecký, Martin Král', Daniel Pekárková, Kristýna Schröder, Felix Integer programming width parameters matroids Graver basis tree-depth fixed parameter tractability 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Marcin Briański and Martin Koutecký and Daniel Král' and Kristýna Pekárková and Felix Schröder 2022 Is Part Of LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2022.29 urn:nbn:de:0030-drops-163702 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.29 eng https://creativecommons.org/licenses/by/4.0/legalcode