A survey on combinatorial optimization

TL;DR

This survey reviews classical and stochastic combinatorial optimization algorithms, introduces new sampling methods, and discusses correlation-robust optimization, providing foundational insights for researchers in the field.

Contribution

It extends classical algorithms to stochastic models, introduces SSA and Boost-and-Sample sampling algorithms, and explores correlation gap concepts for approximation under uncertainty.

Findings

Performance guarantees for SSA and Boost-and-Sample algorithms

Introduction of correlation gap for robust optimization

Framework connecting combinatorial optimization with reinforcement learning

Abstract

This survey revisits classical combinatorial optimization algorithms and extends them to two-stage stochastic models, particularly focusing on client-element problems. We reformulate these problems to optimize element selection under uncertainty and present two key sampling algorithms: SSA and Boost-and-Sample, highlighting their performance guarantees. Additionally, we explore correlation-robust optimization, introducing the concept of the correlation gap, which enables approximations using independent distributions with minimal accuracy loss. This survey analyzes and presents foundational combinatorial optimization methods for researchers at the intersection of this field and reinforcement learning.