A Survey of Exact and Approximation Algorithms for Linear-Parametric Optimization Problems
Levin Nemesch, Stefan Ruzika, Clemens Thielen, Alina Wittmann
TL;DR
This survey surveys structural and algorithmic results for linear-parametric optimization, where a single objective is formed by a linear combination of auxiliary objectives with parameters in $\Lambda$. It highlights fundamental exact methods (Eisner-Severance, Megiddo’s parametric search, minimization diagrams, and lower-image duality) and general approximation frameworks, and then contextualizes these approaches on classic combinatorial problems (shortest path, min-cut/MCF, MST, knapsack, and matching). A central theme is the potential exponential cardinality of optimal solution sets, which motivates parametric approximation and problem-specific strategies that exploit structure (e.g., nesting properties, sparsification). The paper emphasizes deep connections to multi-objective optimization, Lagrangian relaxation, and robust/sensitivity analysis, and points to a rich landscape of results, gaps, and opportunities across single- and multi-parameter settings. Overall, the work distills 128 publications (plus 33 supplements) across 1963–2024, providing both a comprehensive reference and a framework for algorithmic development in parametric 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.