arXiv subjects

Levin Nemesch

Publications and source records attributed to Levin Nemesch.

7 recordsLinked to original sources

arXiv2026

An Adaptive Algorithm for the Approximation of General Linear-Parametric Optimization Problems

Linear-multi-parametric optimization problems are a widely studied class of optimization problems. The objective function in such a problem is affine linear dependent on a parameter vector, and the goal is to compute a set of solutions that contains an optimal solution for every fixed parameter vector. However, this is known to be computationally challenging: The underlying non-parametric problem might be NP-hard, and, in addition, optimal solution sets might have exponential cardinality. Parametric approximation aims at providing polynomial-time algorithms that overcome these challenges. Instead of computing an optimal solution set, the goal is to compute an approximation set that contains only an approximate solution for every fixed parameter vector. Several new parametric approximation algorithms have been developed in recent literature. However, all of these share a common set of assumptions, which limits the class of parametric optimization problems that can be approximated. Namely, they do not allow negative parameter dependencies and have their parameter sets fixed to the positive orthant. We present a new adaptive approximation (and, also, exact) algorithm that can be applied to a wider class of linear-multi-parametric optimization problems. Our algorithm builds upon existing algorithms from both the fields of parametric and multi-objective optimization and generalizes these algorithms. In addition, we provide structural results for the transformation of parameter sets, and demonstrate that, for linear-multi-parametric maximization problems, the assumption of non-negative optimal objective values over the whole parameter set is not sufficient to ensure approximability.

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arXiv2026

Parametric Biobjective Linear Programming

We consider parametric linear programming problems with multiple objective functions depending linearly on some parameter. Both parametric (single-objective) linear programming and (non-parametric) multi-objective linear programming are well-researched topics. However, literature on the combination of both, parametric linear programming with multiple objectives, is scarce. This research gap encourages our work in this field. Our main focus is on biobjective linear programs with a single parameter. We establish a connection of this problem to non-parametric multi-objective problems. Using the so-called weight set decomposition, we are able to explain the behavior of parametric biobjective linear programs when the parameter value is variated. We investigate two special cases of parametric biobjective linear programs: In the first, there is only one parametric objective and, in the second, the parametric dependency is the same for both objectives. We prove that there is a one-to-one correspondence between the solution of the parametric program and the solution of the triobjective program using the weighted sum scalarization. We provide structural insights to the solution of the parametric biobjective linear program with respect to extreme weights of the weight set of the triobjective linear program and develop solution strategies for the parametric program.

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arXiv2026

A High-Performance Parallel Algorithm for Multi-Objective Integer Optimization

Multi-objective integer optimization problems are hard to solve, mainly because the number of nondominated images is often extremely large. We present the first exact algorithm, called PEA, that fully utilizes the multicore architecture of modern hardware. By exploiting the structure of the parameter set of the underlying scalarization, PEA can use a high number of threads while avoiding the usual pitfalls of parallel computing. It is highly scalable and easy to implement. As a result, PEA can solve much larger instances than previous state-of-the-art algorithms. Besides, PEA has a sound theoretical foundation. Unlike other existing parallel algorithms, it always solves the same number of scalarization problems as comparable sequential algorithms. We demonstrate the potential of PEA in a computational study.

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arXiv2025

An Approximation Framework for Parametric Matroid Interdiction Problems

Matroid interdiction problems are well-researched in the field of combinatorial optimization. In the matroid $\ell$-interdiction problem, an interdiction strategy removes a subset of cardinality $\ell$ from the matroid's ground set. The goal is to maximize the weight of a remaining optimal basis. We examine the multi-parametric generalization of this problem, where every weight is given by a linear function depending on a parameter vector. For every parameter value, we are interested in an optimal interdiction strategy and the weight of an optimally interdicted basis. We develop the first framework for lifting approximation algorithms for the non-parametric matroid $\ell$-interdiction problem to its multi-parametric variant. Whenever there exists a $\beta$-approximation algorithm for the non-parametric problem, we obtain an approximation algorithm for the multi-parametric problem with an approximation quality arbitrarily close to $\beta$. Our method yields an FPTAS for partition matroids and a $(1-\varepsilon)\frac{1}{4}$-approximation for graphic matroids. As part of the construction, we develop the first approximation algorithm for a conventional multi-parametric optimization problem in which the parameter vector varies in an arbitrary polytope.

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arXiv2025

A Polynomial-Time Inner Approximation Algorithm for Multi-Objective and Parametric Optimization

In multi-objective optimization, computing the entire non-dominated set (also known as the Pareto front or the Pareto frontier) is often intractable. However, for any multiplicative factor greater than one, an approximation set can be constructed in polynomial time for many problems. In this paper, we use the concept of convex approximation sets: Each point in the non-dominated set is approximated by a convex combination of images of solutions in such a set. Convex approximation sets can be used to efficiently approximate multi-objective optimization problems as well as parametric optimization problems. Recently, Helfrich et al. (2024) presented a convex approximation algorithm that works in an adaptive fashion and runs faster than all previously existing algorithms. We use a different approach for constructing an even more efficient adaptive algorithm for computing convex approximation sets of multi-objective mixed-integer linear programs. Our algorithm is based on the skeleton algorithm for polyhedral inner approximation by Csirmaz (2021). If the weighted sum scalarization can be solved exactly or approximately in polynomial time, our algorithm can find a convex approximation set for an approximation factor arbitrarily close to this solution quality. We demonstrate that our new algorithm runs faster than the current state-of-the-art algorithm from Helfrich et al. (2024) on instances of the multi-objective variants of the assignment problem, the knapsack problem, and the symmetric metric travelling salesman problem.

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arXiv2025

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

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.

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arXiv2022

PaMILO: A Solver for Multi-Objective Mixed Integer Linear Optimization and Beyond

In multi-objective optimization, several potentially conflicting objective functions need to be optimized. Instead of one optimal solution, we look for the set of so called non-dominated solutions. An important subset is the set of non-dominated extreme points. Finding it is a computationally hard problem in general. While solvers for similar problems exist, there are none known for multi-objective mixed integer linear programs (MOMILPs) or multi-objective mixed integer quadratically constrained quadratic programs (MOMIQCQPs). We present PaMILO, the first solver for finding non-dominated extreme points of MOMILPs and MOMIQCQPs. It can be found on github under github.com/FritzBo/PaMILO. PaMILO provides an easy-to-use interface and is implemented in C++17. It solves occurring subproblems employing either CPLEX or Gurobi. PaMILO adapts the Dual-Benson algorithm for multi-objective linear programming (MOLP). As it was previously only defined for MOLPs, we describe how it can be adapted for MOMILPs, MOMIQCQPs and even more problem classes in the future.

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