Hardness Amplification of Optimization Problems.

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 $\Pi$ is direct product feasible if it is possible to efficiently aggregate any $k$ instances of $\Pi$ and form one large instance of $\Pi$ 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 $\Pi$, our hardness amplification theorem may be informally stated as follows: If there is a distribution $\mathcal{D}$ over instances of $\Pi$ of size $n$ such that every randomized algorithm running in time $t(n)$ fails to solve $\Pi$ on $\frac{1}{\alpha(n)}$ fraction of inputs sampled from $\mathcal{D}$, then, assuming some relationships on $\alpha(n)$ and $t(n)$, there is a distribution $\mathcal{D}'$ over instances of $\Pi$ of size $O(n\cdot \alpha(n))$ such that every randomized algorithm running in time $\frac{t(n)}{poly(\alpha(n))}$ fails to solve $\Pi$ on $\frac{99}{100}$ fraction of inputs sampled from $\mathcal{D}'$. 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.