In this paper, we prove a general hardness amplification scheme for optimization problems based on the technique of direct products.

We say that an optimization problem Π is direct product feasible if it is possible to efficiently aggregate any k instances of Π and form one large instance of Π such that given an optimal feasible solution to the larger instance, we can efficiently find optimal feasible solutions to all the k smaller instances. Given a direct product feasible optimization problem Π, our hardness amplification theorem may be informally stated as follows:

If there is a distribution D over instances of Π of size n such that every randomized algorithm running in time t(n) fails to solve Π on 1/α(n) fraction of inputs sampled from D, then, assuming some relationships on α(n) and t(n), there is a distribution D' over instances of Π of size O(n⋅α(n)) such that every randomized algorithm running in time t(n)/poly(α(n)) fails to solve Π on 99/100 fraction of inputs sampled from D'.

As a consequence of the above theorem, we show hardness amplification of problems in various classes such as NP-hard problems like Max-Clique, Knapsack, and Max-SAT, problems in P such as Longest Common Subsequence, Edit Distance, Matrix Multiplication, and even problems in TFNP such as Factoring and computing Nash equilibrium.