TR18-139 Authors: Igor Carboni Oliveira, Rahul Santhanam

Publication: 10th August 2018 22:21

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We show that for several natural problems of interest, complexity lower bounds that are barely non-trivial imply super-polynomial or even exponential lower bounds in strong computational models. We term this phenomenon "hardness magnification". Our examples of hardness magnification include:

1. Let MCSP$[s]$ be the decision problem whose YES instances are truth tables of functions with circuit complexity at most $s(n)$. We show that if MCSP$[2^{\sqrt{n}}]$ cannot be solved on average with zero error by formulas of linear (or even sub-linear) size, then NP does not have polynomial-size formulas. In contrast, Hirahara and Santhanam (2017) showed that MCSP$[2^{\sqrt{n}}]$ cannot be solved in the worst case by formulas of nearly quadratic size.

2. If there is a $c > 0$ such that for each positive integer $d$ there is an $\varepsilon > 0$ such that the problem of checking if an $n$-vertex graph in the adjacency matrix representation has a vertex cover of size $(\log n)^c$ cannot be solved by depth-$d$ AC$^0$ circuits of size $m^{1+\varepsilon}$, where $m = \Theta(n^2)$, then NP does not have polynomial-size formulas.

3. Let $(\alpha, \beta)$-MCSP$[s]$ be the promise problem whose YES instances are truth tables of functions that are $\alpha$-approximable by a circuit of size $s(n)$, and whose NO instances are truth tables of functions that are not $\beta$-approximable by a circuit of size $s(n)$. We show that for any non-trivial choice of the parameters $\alpha$ and $\beta$, if $(\alpha, \beta)$-MCSP$[2^{\sqrt{n}}]$ cannot be solved by randomized algorithms with random access to the input running in sublinear time, then NP is not contained in BPP.

4. If for each probabilistic quasi-linear time machine $M$ using poly-logarithmic many random bits that is claimed to solve Satisfiability, there is a deterministic polynomial-time machine that on infinitely many input lengths $n$ either identifies a satisfiable instance of bitlength $n$ on which $M$ does not accept with high probability or an unsatisfiable instance of bitlength $n$ on which $M$ does not reject with high probability, then NEXP is not contained in BPP.

5. Given functions $s,c \colon N \rightarrow N$ where $s \geq c$, let MKtP$[c,s]$ be the promise problem whose YES instances are strings of Kt complexity (Levin, 1984) at most $c(N)$ and NO instances are strings of Kt complexity greater than $s(N)$. We show that if there is a $\delta > 0$ such that for each $\varepsilon > 0$, MKtP$[N^{\varepsilon}, N^{\varepsilon} + 5 \log(N)]$ requires Boolean circuits of size $N^{1+\delta}$, then EXP is not contained in SIZE(poly).

For each of the cases of magnification above, we observe that standard hardness assumptions imply much stronger lower bounds for these problems than we require for magnification.

We further explore magnification as an avenue to proving strong lower bounds, and argue that magnification circumvents the "natural proofs" barrier of Razborov and Rudich (1997). Examining some standard proof techniques, we find that they fall just short of proving lower bounds via magnification. As one of our main open problems, we ask whether there are other meta-mathematical barriers to proving lower bounds that rule out approaches combining magnification with known techniques.