The Non-hardness of Approximating Circuit Size

Abstract

The Minimum Circuit Size Problem (MCSP) has been the focus of intense study recently; MCSP is hard for SZK under rather powerful reductions [4], and is provably not hard under "local" reductions computable in TIME(n 0.49 ) [26]. The question of whether MCSP is NP-hard (or indeed, hard even for small subclasses of P) under some of

Vaughan Jones, Kolmogorov Complexity, and the New Complexity Landscape around Circuit Minimization

“…Thus one avenue for proving a hardness result for MCSP had seemed to be to improve the hardness result for MKTP, so that it worked for a much larger "gap". This avenue was subsequently blocked, when it was shown that PARITY is not AC 0 -reducible to GapMCSP (or to GapMKTP) for a moderate-sized "gap" [12]. Thus, although it is still open whether MCSP is NP-complete under ≤ AC 0 m reductions, we now know that GapMCSP is not NP-complete under this notion of reducibility.…”

The New Complexity Landscape Around Circuit Minimization

“…Thus one avenue for proving a hardness result for MCSP had seemed to be to improve the hardness result of [6], so that it worked for a much larger "gap". This avenue was subsequently blocked, when it was shown that PARITY is not AC 0 -reducible to GapMCSP (or to GapMKTP) for a moderate-sized "gap" [8]. Thus, although it is still open whether MCSP is NP-complete under ≤ AC 0 m reductions, we now know that GapMCSP is not NP-complete under this notion of reducibility.…”