Amplification and Derandomization Without Slowdown

TL;DR

This paper introduces novel techniques for reducing error probabilities and converting randomized algorithms into deterministic ones without increasing runtime, leveraging connections to bandit problems and graph sparsification.

Contribution

It presents a derandomization method that maintains original runtime, linking derandomization with sketching and sparsification, and demonstrates applications across multiple combinatorial problems.

Findings

Achieved error reduction without slowdown in randomized algorithms.

Established a connection between derandomization and graph sparsification.

Applied techniques successfully to Max-Cut, clique, CSPs, and Reed-Muller decoding.

Abstract

We present techniques for decreasing the error probability of randomized algorithms and for converting randomized algorithms to deterministic (non-uniform) algorithms. Unlike most existing techniques that involve repetition of the randomized algorithm and hence a slowdown, our techniques produce algorithms with a similar run-time to the original randomized algorithms. The amplification technique is related to a certain stochastic multi-armed bandit problem. The derandomization technique - which is the main contribution of this work - points to an intriguing connection between derandomization and sketching/sparsification. We demonstrate the techniques by showing applications to Max-Cut on dense graphs, approximate clique on graphs that contain a large clique, constraint satisfaction problems on dense bipartite graphs and the list decoding to unique decoding problem for the Reed-Muller code.