A Survey of Exact and Approximation Algorithms for Linear-Parametric Optimization Problems

TL;DR

This survey comprehensively reviews exact and approximate algorithms for linear-parametric optimization problems, highlighting structural insights, algorithmic strategies, and connections to multi-objective optimization across various classical problems.

Contribution

It provides a detailed overview of both general strategies and specific algorithms for parametric optimization, integrating recent theoretical and practical developments.

Findings

Coverage of 128 publications from 1963 to 2024

Includes algorithms for shortest path, flow, spanning tree, knapsack, and matching problems

Highlights connections to multi-objective optimization

Abstract

Linear-parametric optimization, where multiple objectives are combined into a single objective using linear combinations with parameters as coefficients, has numerous links to other fields in optimization and a wide range of application areas. In this survey, we provide a comprehensive overview of structural results and algorithmic strategies for solving linear-parametric optimization problems exactly and approximately. Transferring concepts from related areas such as multi-objective optimization provides further relevant results. The survey consists of two parts: First, we list strategies that work in a general fashion and do not rely on specific problem structures. Second, we look at well-studied parametric optimization problems and cover both important theoretical results and specialized algorithmic approaches for these problems. Among these problems are parametric variants of shortest path problems, minimum cost flow and maximum flow problems, spanning tree problems, the knapsack problem, and matching problems. Overall, we cover the results from 128 publications (and refer to 33 supplemental works) published between 1963 and 2024.