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

Figures and Tables from this paper

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

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

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

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.

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

Mach: First order convergence of matroids

  • Eur. J. Comb
  • 2017

A Parameterized Strongly Polynomial Algorithm for Block Structured Integer Programs

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

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

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

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

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

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.