Separations in Proof Complexity and TFNP

It is well-known that Resolution proofs can be efficiently simulated by Sherali–Adams (SA) proofs. We show, however, that any such simulation needs to exploit huge coefficients: Resolution cannot be efficiently simulated by SA when the coefficients are written in unary. We also show that Reversible Resolution (a variant of MaxSAT Resolution) cannot be efficiently simulated by Nullstellensatz (NS).

These results have consequences for total \({\text{\upshape \sffamily NP}} \) search problems. First, we characterise the classes \({\text{\upshape \sffamily PPADS}} \), \({\text{\upshape \sffamily PPAD}} \), \({\text{\upshape \sffamily SOPL}} \) by unary-SA, unary-NS, and Reversible Resolution, respectively. Second, we show that, relative to an oracle, \({\text{\upshape \sffamily PLS}} \not\subseteq {\text{\upshape \sffamily PPP}} \), \({\text{\upshape \sffamily SOPL}} \not\subseteq {\text{\upshape \sffamily PPA}} \), and \({\text{\upshape \sffamily EOPL}} \not\subseteq {\text{\upshape \sffamily UEOPL}} \). In particular, together with prior work, this gives a complete picture of the black-box relationships between all classical \({\text{\upshape \sffamily TFNP}} \) classes introduced in the 1990s.