eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-01-06
1:1
1:13
10.4230/LIPIcs.ITCS.2020.1
article
Hardness Amplification of Optimization Problems
Goldenberg, Elazar
1
Karthik C. S.
2
The Academic College of Tel Aviv Yaffo, Israel
Tel Aviv University, Israel
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol151-itcs2020/LIPIcs.ITCS.2020.1/LIPIcs.ITCS.2020.1.pdf
hardness amplification
average case complexity
direct product
optimization problems
fine-grained complexity
TFNP