Hardness Self-Amplification: Simplified, Optimized, and Unified

Strong (resp. weak) average-case hardness refers to the properties of a computational problem in which a large (resp. small) fraction of instances are hard to solve. We develop a general framework for proving hardness self-amplification, that is, the equivalence between strong and weak average-case hardness. Using this framework, we prove hardness self-amplification for popular problems, such as matrix multiplication, online matrix-vector multiplication, triangle counting of Erdős–Rényi random graphs, and the planted clique problem. As a corollary, we obtain the first search-to-decision reduction for the planted clique problem in a high-error regime. Our framework simplifies, improves, and unifies the previous hardness self-amplification results.