par Aprile, Manuel F. 
;Fiorini, Samuel ;Huynh, Tony ;Joret, Gwenaël ;Wood, David R.
Référence The electronic journal of combinatorics, 28, 4, P4.47
Publication Publié, 2021-10
;Fiorini, Samuel ;Huynh, Tony ;Joret, Gwenaël ;Wood, David R.Référence The electronic journal of combinatorics, 28, 4, P4.47
Publication Publié, 2021-10
Article révisé par les pairs
| Résumé : | Let G be a connected n-vertex graph in a proper minor-closed class G. We prove that the extension complexity of the spanning tree polytope of G is O(n3/2). This improves on the O(n2) bounds following from the work of Wong (1980) and Martin (1991). It also extends a result of Fiorini, Huynh, Joret, and Pashkovich (2017), who obtained a O(n3/2) bound for graphs embedded in a fixed surface. Our proof works more generally for all graph classes admitting strongly sublinear balanced separators: We prove that for every constant β with 0 < β < 1, if G is a graph class closed under induced subgraphs such that all n-vertex graphs in G have balanced separators of size O(nβ ), then the extension complexity of the spanning tree polytope of every connected n-vertex graph in G is O(n1+β ). We in fact give two proofs of this result, one is a direct construction of the extended formulation, the other is via communication protocols. Using the latter approach we also give a short proof of the O(n) bound for planar graphs due to Williams (2002). |
