A note on Probably Certifiably Correct algorithms

TL;DR

This paper discusses algorithms that can solve intractable optimization problems and provide a posteriori certificates of optimality, focusing on probably certifiably correct algorithms with an example for minimum bisection.

Contribution

It introduces the concept of PCC algorithms that not only solve problems but also certify optimality, with a specific fast algorithm for minimum bisection under the stochastic block model.

Findings

Developed a fast PCC algorithm for minimum bisection.

Demonstrated the applicability of PCC algorithms to stochastic block models.

Discussed potential extensions to other optimization problems.

Abstract

Many optimization problems of interest are known to be intractable, and while there are often heuristics that are known to work on typical instances, it is usually not easy to determine a posteriori whether the optimal solution was found. In this short note, we discuss algorithms that not only solve the problem on typical instances, but also provide a posteriori certificates of optimality, probably certifiably correct (PCC) algorithms. As an illustrative example, we present a fast PCC algorithm for minimum bisection under the stochastic block model and briefly discuss other examples.