# 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} }

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.

## 7 Citations

Regular matroids have polynomial extension complexity

- MathematicsMath. Oper. Res.
- 2022

We prove that the extension complexity of the independence polytope of every regular matroid on [Formula: see text] elements is [Formula: see text]. Past results of Wong and Martin on extended…

Extended Formulations for Polytopes of Regular Matroids

- MathematicsArXiv
- 2017

We present a simple proof of the fact that the base (and independence) polytope of a rank $n$ regular matroid over $m$ elements has an extension complexity $O(mn)$.

On 2-Level Polytopes Arising in Combinatorial Settings

- MathematicsSIAM J. Discret. Math.
- 2018

A trade-off formula for the number of cliques and stable sets in a graph; a description of stable matching poly topes as affine projections of certain order polytopes; and a linear-size description of the base polytope of matroids that are 2-level in terms of cuts of an associated tree are presented.

An extended formulation for the 1‐wheel inequalities of the stable set polytope

- MathematicsNetworks
- 2020

The 1‐wheel inequalities for the stable set polytope were introduced by Cheng and Cunningham. In general, there is an exponential number of these inequalities. We present a new polynomial size…

On the Number of Circuits in Regular Matroids (with Connections to Lattices and Codes)

- Computer Science, MathematicsSODA
- 2019

We show that for any regular matroid on $m$ elements and any $\alpha \geq 1$, the number of $\alpha$-minimum circuits, or circuits whose size is at most an $\alpha$-multiple of the minimum size of a…

A note on “A linear‐size zero‐one programming model for the minimum spanning tree problem in planar graphs”

- MathematicsNetworks
- 2019

This note constructs a binary feasible solution to Williams’ formulation that does not represent a spanning tree and restricts the choice of the root vertices in the primal and dual spanning trees, whereas Williams explicitly allowed them to be chosen arbitrarily.

On some problems related to 2-level polytopes

- Mathematics
- 2018

This thesis investigates a number of problems related to 2-level polytopes, in particular regarding their combinatorial structure and extension complexity, and gives an output-efficient algorithm to write down extended formulations for the stable set polytope of perfect graphs.

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