Linear Time Approximation Schemes for the Gale-Berlekamp Game and Related Minimization Problems

TL;DR

This paper introduces linear time approximation schemes for various complex problems including the Gale-Berlekamp Game, NCP, and Unique Games, utilizing a novel technique for handling small objective values, with applications in coding theory and clustering.

Contribution

It presents the first linear time approximation schemes for several dense minimization problems and introduces a new technique for small objective function values.

Findings

Efficient approximation schemes for the Gale-Berlekamp Game and related problems.

Applications to matrix rigidity and error-correcting code decoding.

Linear time algorithms for correlation clustering with fixed clusters.

Abstract

We design a linear time approximation scheme for the Gale-Berlekamp Switching Game and generalize it to a wider class of dense fragile minimization problems including the Nearest Codeword Problem (NCP) and Unique Games Problem. Further applications include, among other things, finding a constrained form of matrix rigidity and maximum likelihood decoding of an error correcting code. As another application of our method we give the first linear time approximation schemes for correlation clustering with a fixed number of clusters and its hierarchical generalization. Our results depend on a new technique for dealing with small objective function values of optimization problems and could be of independent interest.