Rank complexity gap for Lovász-Schrijver and Sherali-Adams proof systems

Abstract

We prove a dichotomy theorem for the rank of the uniformly generated(i.e. expressible in First-Order (FO) Logic) propositional tautologiesin both the Lovász-Schrijver (LS) and Sherali-Adams (SA) proofsystems. More precisely, we first show that the propositional translationsof FO formulae that are universally true, i.e. hold in all finiteand infinite models, have LS proofs whose rank is constant, independentlyfrom the size of the (finite) universe. In contrast to that, we provethat the propositional formulae that hold in all finite models butfail in some infinite structure require proofs whose SA rank grows poly-logarithmically with the size of the universe. Up to now, this kind of so-called "Complexity Gap" theorems have been known for Tree-like Resolution and, in somehow restrictedforms, for the Resolution and Nullstellensatz proof systems. As faras we are aware, this is the first time the Sherali-Adams lift-and-projectmethod has been considered as a propositional proof system. An interesting feature of the SA proof system is that it is static and rank-preserving simulates LS, the Lovász-Schrijver proof system without semidefinitecuts.