# Extended Formulations for Independence Polytopes of Regular Matroids

@article{Kaibel2016ExtendedFF,
title={Extended Formulations for Independence Polytopes of Regular Matroids},
author={Volker Kaibel and Jon Lee and Matthias Walter and Stefan Weltge},
journal={Graphs and Combinatorics},
year={2016},
volume={32},
pages={1931-1944}
}
• Published 15 April 2015
• Mathematics
• Graphs and Combinatorics
We show that the independence polytope of every regular matroid has an extended formulation of size quadratic in the size of its ground set. This generalizes a similar statement for (co-)graphic matroids, which is a simple consequence of Martin’s extended formulation for the spanning-tree polytope. In our construction, we make use of Seymour’s decomposition theorem for regular matroids. As a consequence, the extended formulations can be computed in polynomial time.
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## References

SHOWING 1-10 OF 21 REFERENCES
Submodular Functions, Matroids, and Certain Polyhedra
• J. Edmonds
• Mathematics
Combinatorial Optimization
• 2001
The viewpoint of the subject of matroids, and related areas of lattice theory, has always been, in one way or another, abstraction of algebraic dependence or, equivalently, abstraction of the
Decomposition of regular matroids
The matching polytope has exponential extension complexity
By a known reduction this also improves the lower bound on the extension complexity for the TSP polytope from 2Ω(√n) to 2 Ω(n).
Theta rank, levelness, and matroid minors
• Mathematics
J. Comb. Theory, Ser. B
• 2017
An algorithm for determining whether a given binary matroid is graphic.
1. Introduction. In a recent series of papers [l-4] on graphs and matroids I used definitions equivalent to the following. A binary chain-group N on a. finite set M is a class of subsets of M forming
Extended formulations for sparsity matroids
• Mathematics
Math. Program.
• 2016
A polynomial-size extended formulation for the base polytope of a k-sparsity matroid using the technique developed by Faenza et al. recently that uses a randomized communication protocol is shown.
Combinatorial optimization. Polyhedra and efficiency.
This book shows the combinatorial optimization polyhedra and efficiency as your friend in spending the time in reading a book.
A Short Proof that the Extension Complexity of the Correlation Polytope Grows Exponentially
• Mathematics
Discret. Comput. Geom.
• 2015
The main innovative aspect of the proof is a simple combinatorial argument showing that the rectangle covering number of the unique-disjointness matrix is at least $$1.5^n$$1.58n, which slightly improves on the previously best known lower bounds $1.24n and$.31n, respectively.