文章基本信息

标题：Matrices of Optimal Tree-Depth and Row-Invariant Parameterized Algorithm for Integer Programming
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作者：Timothy F. N. Chan ; Jacob W. Cooper ; Martin Kouteck{'y 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2020
卷号：168
页码：26:1-26:19
DOI：10.4230/LIPIcs.ICALP.2020.26
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：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 tree-depth d and largest entry Î" are solvable in time g(d,Î") poly(n) for some function g, i.e., fixed parameter tractable when parameterized by tree-depth d and Î". However, the tree-depth of a constraint matrix depends on the positions of its non-zero entries and thus does not reflect its geometric structure. In particular, 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. We prove that the branch-depth of the matroid defined by the columns of the constraint matrix is equal to the minimum tree-depth of a row-equivalent matrix. We also design a fixed parameter algorithm parameterized by an integer d and the entry complexity of an input matrix that either outputs a matrix with the smallest dual tree-depth that is row-equivalent to the input matrix or outputs that there is no matrix with dual tree-depth at most d that is row-equivalent to the input matrix. Finally, we use these results to obtain a fixed parameter algorithm for integer programming parameterized by the branch-depth of the input constraint matrix and the entry complexity. The parameterization by branch-depth cannot be replaced by the more permissive notion of branch-width.
关键词：Matroid algorithms; width parameters; integer programming; fixed parameter tractability; branch-width; branch-depth