Nearly Optimal Average-Case Complexity of Counting Bicliques Under SETH

Abstract

In this paper, we seek a natural problem and a natural distribution of instances such that any O(n c− ) 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 and a and b are constants. Our distribution consists of the binomial random bipartite graphs B αn,βn with edge density 1/2, where α and β are drawn uniformly at random from {1, . . . , a} and {1, . . . , b}, respectively. We determine the nearly optimal average-case complexity of this counting problem by proving the following results.