Hardness Magnification for all Sparse NP Languages

Abstract

In the Minimum Circuit Size Problem (MCSP[s(m)]), we ask if there is a circuit of size s(m) computing a given truth-table of length n = 2 m . Recently, a surprising phenomenon termed as hardness magnification by [Oliveira and Santhanam, FOCS 2018] was discovered for MCSP[s(m)] and the related problem MKtP of computing time-bounded Kolmogorov complexity. In [Oliveira and Santhanam, FOCS 2018], [Oliveira, Pich, and Santhanam, CCC 2019], and Williams, STOC 2019], it was shown that minor (n 1+ε -style) lower bounds for MCSP [2 o(m) ] or MKtP [2 o(m) ] would imply breakthrough circuit lower bounds such as NP ⊂ P /poly , NP ⊂ NC 1 , or EXP ⊂ P /poly .We consider the question: What is so special about MCSP and MKtP? Why do they admit this striking phenomenon? One simple property is that all variants of MCSP (and MKtP) considered in prior work are sparse languages. For example,-sparse. We show that there is a hardness magnification phenomenon for all equally-sparse NP languages. Formally, suppose there is an ε > 0 and a language L ∈ NP which is-sparse, and L / ∈ Circuit[n 1+ε ]. Then NP does not have n k -size circuits for all k. We prove analogous theorems for De Morgan formulas, B 2 -formulas, branching programs, AC 0 [6] and TC 0 circuits, and more: improving the state of the art in NP lower bounds against any of these models by an ε factor in the exponent would already imply NP lower bounds for all fixed polynomials. In fact, in our proofs it is not necessary to prove a (say) n 1+ε circuit size lower bound for L: one only has to prove a lower bound against n 1+ε -time n ε -space deterministic algorithms with n ε advice bits. Such lower bounds are well-known for non-sparse problems.Building on our techniques, we also show interesting new hardness magnifications for search-MCSP and search-MKtP (where one must output small circuits or short representations of strings), showing consequences such as ⊕P (or PP, PSPACE, and EXP) is not contained in P /poly (or NC 1 , AC 0 [6], or branching programs of polynomial size). For instance, if there is an ε > 0 such that search-MCSP[2 βm ] does not have De Morgan formulas of size n 3+ε for all constants β > 0, then ⊕P ⊂ NC 1 .

Sharp threshold results for computational complexity

“…Several recent papers [19][20][21][53][54][55]] also establish hardness magnification results for problems such as MCSP and MKtP. Most recently, the reference [19] proves Theorem 1.3 for deterministic formula lower bounds of size n 3+ε ; we generalize their results for probabilistic formula lower bounds of size n 2+ε . Our Theorem 1.5 generalizes n 2−ε -size formula lower bounds for MCSP [32,54,55] on larger circuit sizes.…”

“…See Section 2.2 for formal definitions. Theorem 1.3 (From "Weak" Lower Bounds to Super-Polynomial Lower Bounds, adapting [19]).…”

“…Our Theorem 1.5 generalizes n 2−ε -size formula lower bounds for MCSP [32,54,55] on larger circuit sizes. In [19,54,55] it is shown that an n 3+ε -size formula lower bound for MKtP[n o (1) ] implies EXP NC 1 , and [32,54] showed that MKtP[polylog(n)] does not have n 2−ε -size formulas. That is, the "gap" between the magnification threshold and known lower bounds was n 3+ε versus n 2−ε .…”

“…We first establish the magnification theorems, which are an adaptation of the main results of [19]. We need the following theorem from [19]. log n ≤ log(S sparse ) ≤ n 1−Ω (1) .…”

The New Complexity Landscape Around Circuit Minimization

“…It is important to note, in this regard, that lower bounds have been proved for MCSP that essentially match the strongest lower bounds that we have for any problems in NP [16]. There is now a significant body of work, showing that slight improvements to those bounds, or other seemingly-attainable lower bounds for GapMKtP or GapMCSP or related problems, would yield dramatic complexity class separations [12][13][14][15]34,38,39,45].…”