Separations in Proof Complexity and TFNP

It is well-known that Resolution proofs can be efficiently simulated by Sherali-Adams (SA) proofs. We show 1, 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 NP search problems. First, we characterise the classes PPADS, PPAD, SOPL by unary-SA, unary-NS, and Reversible Resolution, respectively. Second, we show that, relative to an oracle, PLS $\nsubseteq$ PPP, SOPL $\nsubseteq$ PPA, and EOPL $\nsubseteq$ UEOPL. In particular, together with prior work, this gives a complete picture of the black-box relationships between all classical TFNP classes introduced in the 1990s. 1 This is an extended abstract. For the full version of this article, please refer to [GHJ+22b].

Authors

• Mika Göös
• Alexandros Hollender
• Siddhartha Jain 0002
• Gilbert Maystre
• William Pires
• Robert Robere
• Ran Tao 0013

Keywords

• Computer science
• Closed box
• Search problems
• Complexity theory
• Proof Complexity
• Search Problem
• Total Problems
• Classical Classes
• Total Search
• Inactive
• Decision Tree
• Large Degree
• Directed Graph
• Proof Of Proposition
• Out-degree
• Complex Communication
• Monomial
• Boolean Variable
• Grid Nodes
• Multiset
• Distinct Nodes
• Proper Source