Nearly Optimal Average-Case Complexity of Counting Bicliques Under SETH

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.