Samuel Fiorini

Exponential Lower Bounds for Polytopes in Combinatorial Optimization

We solve a 20-year old problem posed by Yannakakis and prove that no polynomial-size linear program (LP) exists whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope and the stable set polytope. These results were discovered through a new connection that we make between one-way quantum communication protocols and semidefinite programming reformulations of LPs.

Approximation Limits of Linear Programs (Beyond Hierarchies)

We develop a framework for proving approximation limits of polynomial size linear programs (LPs) from lower bounds on the nonnegative ranks of suitably defined matrices. This framework yields unconditional impossibility results that are applicable to any LP as opposed to only programs generated by hierarchies. Using our framework, we prove that O(n1/2-ϵ)-approximations for CLIQUE require LPs of size 2nΩ(ϵ). This lower bound applies to LPs using a certain encoding of CLIQUE as a linear optimization problem. Moreover, we establish a similar result for approximations of semidefinite programs by LPs. Our main technical ingredient is a quantitative improvement of Razborov’s [38] rectangle corruption lemma for the high error regime, which gives strong lower bounds on the nonnegative rank of shifts of the unique disjointness matrix.

Extended formulations, nonnegative factorizations, and randomized communication protocols

An extended formulation of a polyhedron

Planar graph bipartization in linear time

P

Minimum Entropy Coloring

P is a linear description of a polyhedron

0, 1/2-Cuts and the Linear Ordering Problem: Surfaces That Define Facets

Q