The Minimum Circuit Size Problem (MCSP) asks to determine the minimum size of a circuit computing a given truth table. MCSP is a natural and powerful string compression problem whose NPhardness remains open. Recently, Oliveira and Santhanam [FOCS 2018] and Oliveira, Pich, and Santhanam [ECCC 2018] demonstrated a "hardness magnification" phenomenon for MCSP in restricted settings. Letting MCSP[s(n)] be the problem of deciding if a truth table of length 2 n has circuit complexity at most s(n), they proved that small (fixed-polynomial) average case circuit/formula lower bounds for MCSP[2 √ n ], or lower bounds for approximating * Supported by NSF CAREER 1741615. † Portions of this work were completed while the author was a PhD student at MIT, and as a Research Fellow at the Simons Institute, UC Berkeley.
We prove that if every problem in NP has n k -size circuits for a fixed constant k, then for every NPverifier and every yes-instance x of length n for that verifier, the verifier's search space has an n O(k 3 ) -size witness circuit: a witness for x that can be encoded with a circuit of only n O(k 3 ) size. An analogous statement is proved for nondeterministic quasi-polynomial time, i.e., NQP = NTIME[n log O( 1) n ]. This significantly extends the Easy Witness Lemma of Impagliazzo, Kabanets, and Wigderson [JCSS'02] which only held for larger nondeterministic classes such as NEXP.As a consequence, the connections between circuit-analysis algorithms and circuit lower bounds can be considerably sharpened: algorithms for approximately counting satisfying assignments to given circuits which improve over exhaustive search can imply circuit lower bounds for functions in NQP, or even NP. To illustrate, applying known algorithms for satisfiability of ACC • THR circuits [R. Williams, STOC 2014] we conclude that for every fixed k, NQP does not have n log k n -size ACC • THR circuits.
We prove that if every problem in \sansN \sansP has n k -size circuits for a fixed constant k, then for every \sansN \sansP -verifier and every yes-instance x of length n for that verifier, the verifier's search space has an n O(k 3 ) -size witness circuit: A witness for x that can be encoded with a circuit of only n O(k 3 ) size. An analogous statement is proved for nondeterministic quasi-polynomial time, i.e., \sansN \sansQ \sansP = \sansN \sansT \sansI \sansM \sansE [n log O(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 \sansN \sansE \sansX \sansP . 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 \sansN \sansQ \sansP , or even \sansN \sansP . To illustrate, applying known algorithms for satisfiability of \sansA \sansC \sansC \circ \sansT \sansH \sansR 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, \sansN \sansQ \sansP does not have n log k nsize \sansA \sansC \sansC \circ \sansT \sansH \sansR circuits.
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