Extended formulations for sparsity matroids

We show the existence of a polynomial-size extended formulation for the base polytope of a $$(k,\ell )$$(k,ℓ)-sparsity matroid. For an undirected graph $$G=(V,E)$$G=(V,E), the size of the formulation is $$O(|V|\cdot |E|)$$O(|V|·|E|) when $$k \ge \ell $$k≥ℓ and $$O(|V|^2 |E|)$$O(|V|2|E|) when $$k \le \ell $$k≤ℓ. To this end, we employ the technique developed by Faenza et al. recently that uses a randomized communication protocol.