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. 
Regular matroids have polynomial extension complexity
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
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
TLDR
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
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)
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”
TLDR
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
TLDR
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.

References

SHOWING 1-10 OF 21 REFERENCES
Subgraph polytopes and independence polytopes of count matroids
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
TLDR
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).
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
TLDR
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.
TLDR
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
TLDR
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.
...
...