Regular matroids have polynomial extension complexity

TL;DR

This paper proves that the independence polytope of any regular matroid has polynomial extension complexity, specifically O(n^6), extending previous results for special cases like graphic and co-graphic matroids.

Contribution

It establishes a polynomial upper bound on the extension complexity for the independence polytope of all regular matroids, a previously open problem.

Findings

Extension complexity of independence polytope is O(n^6) for regular matroids.

Special case of graphic and co-graphic matroids has O(n^2) extension complexity.

Extension complexity of circuit dominants of regular matroids is also O(n^2).

Abstract

We prove that the extension complexity of the independence polytope of every regular matroid on $n$ elements is $O(n^6)$. Past results of Wong and Martin on extended formulations of the spanning tree polytope of a graph imply a $O(n^2)$ bound for the special case of (co)graphic matroids. However, the case of a general regular matroid was open, despite recent attempts. We also consider the extension complexity of circuit dominants of regular matroids, for which we give a $O(n^2)$ bound.