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

@inproceedings{Chan2020MatricesOO, title={Matrices of Optimal Tree-Depth and Row-Invariant Parameterized Algorithm for Integer Programming}, author={Timothy F. N. Chan and Jacob W. Cooper and Martin Kouteck{\'y} and Daniel Kr{\'a}l and Krist{\'y}na Pek{\'a}rkov{\'a}}, booktitle={ICALP}, year={2020} }

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 Δ are solvable in time g(d,Δ) poly(n) for some function g, i.e., fixed parameter tractable when parameterized by tree-depth d and Δ. 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…

## 4 Citations

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

- Computer ScienceICALP
- 2022

This work designs 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 and provides structural results characterizing the existence of a sparse equivalent matrix in terms of the structural properties of the associated column matroid.

### Sometimes, Convex Separable Optimization Is Much Harder than Linear Optimization, and Other Surprises

- Mathematics, Computer ScienceArXiv
- 2021

This work proves that separable convex optimization is much harder than linear optimization, and gives the first non-trivial lower and upper bounds on the norm of mixed Graver basis elements.

### Resolving Infeasibility of Linear Systems: A Parameterized Approach

- Mathematics, Computer ScienceIPEC
- 2019

These algorithms capture the case of Directed Feedback Arc Set, a fundamental parameterized problem whose parameterized tractability was shown by Chen et al. (STOC 2008) and establish parameterized intractability results even in very restricted settings.

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