# 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}
}
• Published in ICALP 15 July 2019
• Computer Science
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

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## References

SHOWING 1-10 OF 42 REFERENCES

### Minkowski's Convex Body Theorem and Integer Programming

• R. Kannan
• Mathematics, Computer Science
Math. Oper. Res.
• 1987
An algorithm for solving Integer Programming problems whose running time depends on the number n of variables as nOn by reducing an n variable problem to 2n5i/2 problems in n-i variables for some i greater than zero chosen by the algorithm.

### On integer programming and the branchwidth of the constraint matrix, Integer Programming and Combinatorial Optimization

• 12th International IPCO Conference
• 2007

• Eur. J. Comb
• 2017

### A Parameterized Strongly Polynomial Algorithm for Block Structured Integer Programs

• Computer Science
ICALP
• 2018
It is shown that ILP is FPT parameterized by the largest coefficient $\|A\|_\infty$ and the primal or dual treedepth of $A$, and that this parameterization cannot be relaxed, signifying substantial progress in understanding the parameterized complexity of ILP.

### Rearranging Matrices to Block-Angular form for Decomposition (And Other) Algorithms

• Computer Science
• 1971
This article presents a systematic method for effecting a block-angular permutation of arbitrary coefficient matrix of large numerical problems, and the results of manipulations of matrices with more than 300 rows and 2500 columns are shown.

### An Algorithmic Theory of Integer Programming

• Computer Science, Mathematics
ArXiv
• 2019
It is shown that integer programming can be solved in time, and a strongly-polynomial algorithm is derived, that is, with running time $g(a,d)\textrm{poly}(n)$, independent of the rest of the input data.

### Faster Algorithms for Integer Programs with Block Structure

• Computer Science, Mathematics
ICALP
• 2018
The algorithm is based on a new upper bound on the $l_1$-norm of an element of the "Graver basis" of an integer matrix and on a proximity bound between the LP and IP optimal solutions tailored for IPs with block structure.

### Covering a tree with rooted subtrees - parameterized and approximation algorithms

• Computer Science
SODA
• 2018
This work considers the multiple traveling salesman problem on a weighted tree, and shows that an ILP with such a structure is FPT, which is a generalization of an earlier FPT result for n-fold integer programming by Hemmecke, Onn and Romanchuk.

### Deciding First Order Properties of Matroids

• Mathematics
ICALP
• 2012
It is shown that the problem of deciding the existence of a circuit of length at most k containing two given elements is fixed parameter tractable for regular matroids.