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TR19-118 | 5th September 2019 19:02

Hardness Magnification for all Sparse NP Languages


Authors: Lijie Chen, Ce Jin, Ryan Williams
Publication: 15th September 2019 17:51
Downloads: 1711


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 [McKay, Murray, and Williams, STOC 2019], it was shown that minor (n^{1+eps}-style) lower bounds for MCSP[2^o(m)] or MKtP[2^o(m)] would imply breakthrough circuit lower bounds such as NP is not in P/poly, NP is not in NC^1, or EXP is not in 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, MCSP[s(m)] has 2^{O(s(m))} yes-instances of length n=2^m, so MCSP[2^o(m)] is 2^{n^o(1)}-sparse.

We show that there is a hardness magnification phenomenon for all equally-sparse NP languages. Formally, suppose there is an eps > 0 and a language L in NP which is 2^{n^o(1)}-sparse, and L is not in Circuit[n^{1+eps}]. 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 eps 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+eps} circuit size lower bound for L: one only has to prove a lower bound against n^{1+eps}-time n^eps-space deterministic algorithms with n^eps 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 Parity-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 eps > 0 such that search-MCSP[2^{beta m}] does not have De Morgan formulas of size n^{3+eps} for all constants beta > 0, then Parity-P is not in NC^1.

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