Circuit Lower Bounds for Nondeterministic Quasi-polytime from a New Easy Witness Lemma

Abstract

© 2020 Society for Industrial and Applied Mathematics. We prove that if every problem in N P has nk-size circuits for a fixed constant k, then for every N P -verifier and every yes-instance x of length n for that verifier, the verifier's search space has an nO(k3)-size witness circuit: A witness for x that can be encoded with a circuit of only nO(k3) size. An analogous statement is proved for nondeterministic quasi-polynomial time, i.e., N Q P = N T I M E [nlogO(1) n]. This significantly extends the Easy Witness Lemma of Impagliazzo, Kabanets, and Wigderson [J. Comput. System Sci., 65 (2002), pp. 672-694] which only held for larger nondeterministic classes such as N E X P . As a consequence, the connections between circuit-analysis algorithms and circuit lower bounds can be considerably sharpened: Algorithms for approximately counting satisfying assignments for given circuits which improve over exhaustive search can imply circuit lower bounds for functions in N Q P , or even N P . To illustrate, applying known algorithms for satisfiability of A C C T H R circuits [R. Williams, New algorithms and lower bounds for circuits with linear threshold gates, in Proceedings of the 46th Annual ACM Symposium on Theory of Computing, ACM, New York, 2014, pp. 194-202] we conclude that for every fixed k, N Q P does not have nlogk nsize A C C T H R circuits.