Some 0/1 polytopes need exponential size extended formulations

TL;DR

This paper demonstrates that certain 0/1 polytopes, including TSP polytopes, require exponential size extended formulations, indicating fundamental complexity limitations for linear programming approaches.

Contribution

It proves exponential lower bounds on the extension complexity of specific 0/1 polytopes, including matroid and TSP polytopes, under standard complexity assumptions.

Findings

Existence of 0/1 polytopes with exponential extension complexity

Extension complexity of certain polytopes is at least 2^{n/2*(1-o(1))}

Results imply no compact LP formulation for TSP polytope under common complexity assumptions

Abstract

We prove that there are 0/1 polytopes P that do not admit a compact LP formulation. More precisely we show that for every n there is a sets X \subseteq {0,1}^n such that conv(X) must have extension complexity at least 2^{n/2 * (1-o(1))}. In other words, every polyhedron Q that can be linearly projected on conv(X) must have exponentially many facets. In fact, the same result also applies if conv(X) is restricted to be a matroid polytope. Conditioning on NP not contained in P_{/poly}, our result rules out the existence of any compact formulation for the TSP polytope, even if the formulation may contain arbitrary real numbers.