On Semi-Algebraic Proofs and Algorithms.

We give a new characterization of the Sherali-Adams proof system, showing that there is a degree-d Sherali-Adams refutation of an unsatisfiable CNF formula C if and only if there is an ε > 0 and a degree-d conical junta J such that violC(x) − ε = J , where violC(x) counts the number of falsified clauses of C on an input x. Using this result we show that the linear separation complexity, a complexity measure recently studied by Hrubeš (and independently by de Oliveira Oliveira and Pudlák under the name of weak monotone linear programming gates), monotone feasibly interpolates Sherali-Adams proofs. We then investigate separation results for violC(x) − ε. In particular, we give a family of unsatisfiable CNF formulas C which have polynomial-size and small-width resolution proofs, but for which any representation of violC(x)− 1 by a conical junta requires degree Ω(n); this resolves an open question of Filmus, Mahajan, Sood, and Vinyals. Since Sherali-Adams can simulate resolution, this separates the non-negative degree of violC(x) − 1 and violC(x) − ε for arbitrarily small ε > 0. Finally, by applying lifting theorems, we translate this lower bound into new separation results between extension complexity and monotone circuit complexity. Research supported by NSERC. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 3 (2022)