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In this paper, we seek a natural problem and a natural distribution ofinstances such that any $O(n^{c-\epsilon})$-time algorithm fails to solve mostinstances 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 provedthat 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 onrandom bipartite graph for any constant $\epsilon>0$ under the StrongExponential Time Hypotheis. Then, we amplify the hardness of this problem usingthe direct product theorem and Yao's XOR lemma by presenting a generalframework of hardness amplification in the setting of fine-grained complexity.

Nearly Optimal Average-Case Complexity of Counting Bicliques Under SETH