Revision #1 Authors: Shuichi Hirahara

Accepted on: 28th February 2019 09:41

Downloads: 359

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There are significant obstacles to establishing an equivalence between the worst-case and average-case hardness of NP: Several results suggest that black-box worst-case to average-case reductions are not likely to be used for reducing any worst-case problem outside coNP to a distributional NP problem.

This paper overcomes the barrier. We present the first non-black-box worst-case to average-case reduction from a problem conjectured to be outside coNP to a distributional NP problem. Specifically, we consider the minimum time-bounded Kolmogorov complexity problem (MINKT), and prove that there exists a zero-error randomized polynomial-time algorithm approximating the minimum time-bounded Kolmogorov complexity $k$ within an additive error $\widetilde{O}(\sqrt{k})$ if its average-case version admits an errorless heuristic polynomial-time algorithm. We observe that the approximation version of MINKT is Random 3SAT-hard, and more generally it is harder than avoiding any polynomial-time computable hitting set generator that extends its seed of length $n$ by $\widetilde{\omega}(\sqrt{n})$, which provides strong evidence that the approximation problem is outside coNP and thus our reductions are non-black-box. Our reduction can be derandomized at the cost of the quality of the approximation. We also show that, given a truth table of size $2^n$, approximating the minimum circuit size within a factor of $2^{(1 - \epsilon) n}$ is in BPP for some constant $\epsilon > 0$ if and only if its average-case version is easy.

Our results can be seen as a new approach for excluding Heuristica. In particular, proving NP-hardness of the approximation versions of MINKT or the Minimum Circuit Size Problem (MCSP) is sufficient for establishing an equivalence between the worst-case and average-case hardness of NP.

The presentation of the results is improved (see Figure 2). Added derandomized versions of the reductions.

There are significant obstacles to establishing an equivalence between the worst-case and average-case hardness of NP: Several results suggest that black-box worst-case to average-case reductions are not likely to be used for reducing any worst-case problem outside coNP to a distributional NP problem.

This paper overcomes the barrier. We present the first non-black-box worst-case to average-case reduction from a problem outside coNP (unless Random 3SAT is easy for coNP algorithms) to a distributional NP problem. Specifically, we consider the minimum time-bounded Kolmogorov complexity problem (MINKT), and prove that there exists a zero-error randomized polynomial-time algorithm approximating the minimum time-bounded Kolmogorov complexity $k$ within an additive error $\widetilde{O}(\sqrt{k})$ if its average-case version admits an errorless heuristic polynomial-time algorithm. (The converse direction also holds under a plausible derandomization assumption.) We also show that, given a truth table of size $2^n$, approximating the minimum circuit size within a factor of $2^{(1 - \epsilon) n}$ is in BPP for some constant $\epsilon > 0$ if and only if its average-case version is easy.

Based on our results, we propose a research program for excluding Heuristica, i.e., establishing an equivalence between the worst-case and average-case hardness of NP through the lens of MINKT or the Minimum Circuit Size Problem (MCSP).