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Revision #1 to TR18-138 | 28th February 2019 09:41

#### Non-black-box Worst-case to Average-case Reductions within NP

Revision #1
Authors: Shuichi Hirahara
Accepted on: 28th February 2019 09:41
Keywords:

Abstract:

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.

Changes to previous version:

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

### Paper:

TR18-138 | 10th August 2018 12:39

#### Non-black-box Worst-case to Average-case Reductions within NP

TR18-138
Authors: Shuichi Hirahara
Publication: 10th August 2018 13:07
Keywords:

Abstract:

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).

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