In this paper, we seek a natural problem and a natural distribution of instances such that any $O(n^{c-\epsilon})$-time algorithm fails to solve most instances drawn from the distribution, while the problem admits an $n^{c+o(1)}$-time algorithm that correctly solves all instances. Specifically, we consider the $K_{a,b}$ counting problem in a random bipartite graph, where $K_{a,b}$ is a complete bipartite graph for constants $a$ and $b$. We proved that the $K_{a,b}$ counting problem admits an $n^{a+o(1)}$-time algorithm if $a\geq 8$, while any $n^{a-\epsilon}$-time algorithm fails to solve it even on random bipartite graph for any constant $\epsilon>0$ under the Strong Exponential Time Hypotheis. Then, we amplify the hardness of this problem using the direct product theorem and Yao's XOR lemma by presenting a general framework of hardness amplification in the setting of fine-grained complexity.

中文翻译：

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SETH 下计算 Bicliques 的近乎最优的平均情况复杂度

在本文中，我们寻求一个自然问题和实例的自然分布，使得任何 $O(n^{c-\epsilon})$-time 算法都无法解决从分布中提取的大多数实例，而问题承认 $ n^{c+o(1)}$-time 算法可以正确解决所有实例。具体来说，我们考虑随机二部图中的 $K_{a,b}$ 计数问题，其中 $K_{a,b}$ 是常量 $a$ 和 $b$ 的完整二部图。我们证明了 $K_{a,b}$ 计数问题承认 $n^{a+o(1)}$-time 算法如果 $a\geq 8$，而任何 $n^{a-\epsilon}在强指数时间假设下，即使在随机二部图上，对于任何常数 $\epsilon>0$，$-time 算法也无法解决它。然后，我们使用直积定理和Yao'来放大这个问题的难度