On the Combinatorial Lower Bound for the Extension Complexity of the Spanning Tree Polytope
Abstract
In the study of extensions of polytopes of combinatorial optimization problems, a notorious open question is that for the size of the smallest extended formulation of the Minimum Spanning Tree problem on a complete graph with $n$ nodes. The best known lower bound is the trival (dimension) bound, $\Omega(n^2)$, the best known upper bound is the extended formulation by Wong (1980) of size $O(n^3)$ (also Martin, 1991). In this note we give a nondeterministic communication protocol with cost $\log_2(n^2\log n)+O(1)$ for the support of the spanning tree slack matrix. This means that the combinatorial lower bounds can improve the trivial lower bound only by a factor of (at most) $O(\log n)$.
- Publication:
-
arXiv e-prints
- Pub Date:
- February 2017
- DOI:
- 10.48550/arXiv.1702.01424
- arXiv:
- arXiv:1702.01424
- Bibcode:
- 2017arXiv170201424K
- Keywords:
-
- Computer Science - Discrete Mathematics;
- Mathematics - Optimization and Control
- E-Print:
- 9p