Sherali-Adams and the binary encoding of combinatorial principles

TL;DR

This paper investigates the Sherali-Adams system with binary encodings of combinatorial principles, revealing exponential size lower bounds and contrasting behaviors with other encoding schemes.

Contribution

It establishes exponential lower bounds for binary-encoded Pigeonhole Principle in Sherali-Adams and compares rank requirements across encoding methods.

Findings

Binary encoding of Pigeonhole Principle requires exponential-sized SA refutations.

Binary encoding of Least Number Principle admits polynomial-sized SA refutations with logarithmic rank.

Between SA and Lasserre, the Least Number Principle needs linear rank, while Pigeonhole becomes constant rank.

Abstract

We consider the Sherali-Adams (SA) refutation system together with the unusual binary encoding of certain combinatorial principles. For the unary encoding of the Pigeonhole Principle and the Least Number Principle, it is known that linear rank is required for refutations in SA, although both admit refutations of polynomial size. We prove that the binary encoding of the Pigeonhole Principle requires exponentially-sized SA refutations, whereas the binary encoding of the Least Number Principle admits logarithmic rank, polynomially-sized SA refutations. We continue by considering a refutation system between SA and Lasserre (Sum-of-Squares). In this system, the Least Number Principle requires linear rank while the Pigeonhole Principle becomes constant rank.