arXiv subjects

Clemens Thielen

Publications and source records attributed to Clemens Thielen.

At least 19 recordsLinked to original sources

arXiv2026

A Multi-Objective Approach to Curriculum-Based Course Timetabling with Continuity Across Semesters

We study a curriculum-based university course timetabling problem in which the preferences of two key stakeholder groups - lecturers and students - must be balanced while maintaining continuity across semesters in a weekly repeating timetable. While existing approaches typically rely on single-objective formulations or aggregate multiple objectives into a weighted sum, this can obscure the underlying trade-offs between conflicting stakeholder preferences. We therefore propose a multi-objective mixed-integer programming approach that explicitly separates lecturer and student objectives and incorporates timetable continuity by limiting the number of changes, called perturbations, in the time period assignments of selected courses relative to the corresponding semester of the previous academic year. To explore the resulting trade-offs, we develop a multi-objective solution approach based on the lexicographic $\varepsilon$-constraint method, enabling the computation of a representative set of solutions whose images, i.e., their vectors of objective values, cover different regions of the objective space. The approach is evaluated on real-world instances from the Straubing Campus of the Technical University of Munich. The computational results reveal a clear and consistent trade-off between lecturers' and students' objectives across all instances. Moreover, the number of allowed perturbations is identified as a key decision parameter: relaxing this constraint significantly improves timetable quality for both stakeholder groups, although diminishing returns are observed beyond certain thresholds. Overall, the proposed approach provides decision support by generating a diverse set of optimized timetables and enabling a transparent analysis of stakeholder trade-offs and continuity for the practical timetable planning process.

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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

Label Correcting Algorithms for the Multiobjective Temporal Shortest Path Problem

Given a directed, discrete-time temporal graph $G=(V,R)$, a start node $s\in V$, and $p\geq1$ objectives, the single-source multiobjective temporal shortest path problem asks, for each $v\in V$, for the set of nondominated images of temporal $s$-$v$-paths together with a corresponding efficient path for each image. A recent general label setting algorithm for this problem relies on two properties of the objectives - monotonicity and isotonicity. Monotonicity generalizes the nonnegativity assumption required by label setting methods for the classical additive single-objective shortest path problem on static graphs, while isotonicity ensures that the order of the objective values of two paths is preserved when both are extended by the same arc. In this paper, we study the problem without assuming monotonicity and/or isotonicity. A key difficulty in this setting is that zero-duration temporal cycles may need to be traversed an arbitrary finite number of times to generate all nondominated images. This motivates the study of a restricted problem variant in which a maximum admissible path length $K$ is imposed, and only paths containing at most $K$ arcs are considered. We develop general label correcting algorithms for this setting and establish several sufficient conditions under which such a bound is not required, implying that the algorithms compute all nondominated images.

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arXiv2026

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 General Framework for Physician Rostering Using Mixed-Integer Programming and a Web-Based Graphical User Interface

Physician rostering in hospitals is complex due to varying shift structures, qualifications, and department- or hospital-specific regulations. Most existing optimization models are highly tailored to a single hospital or department and rarely see practical use. We present a general framework and a corresponding mixed-integer programming (MIP) model for physician rostering that accommodates a wide variety of roster structures and constraints. The model is integrated into a web application with an advanced graphical user interface (GUI), allowing physicians to specify preferences and hospital staff to configure the MIP model to their roster requirements without any mathematical or technical background. This approach enables easy adaptation to different hospitals or departments and straightforward updates in response to structural changes, such as new duties or modified qualifications. The applicability and effectiveness of the framework are demonstrated using real-world data from three departments in different hospitals specializing in internal medicine, cardiology, and orthopedics/trauma surgery. In one department, the system is already in everyday use, while in the other two, our model achieves comparable or improved roster quality relative to existing department-specific models, highlighting its potential as a versatile and practical tool for physician rostering.

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arXiv2025

A fast approximate column-and-constraint generation method for two-stage robust mixed-integer programs

This paper presents a new column-and-constraint generation method for two-stage robust mixed-integer programs with finite uncertainty sets. Our method combines and extends speed-up techniques used in previous column-and-constraint generation methods and introduces several new techniques. In particular, it uses dual bounds for second-stage problems in order to allow a faster identification of the next promising scenario to be added to the master problem. Moreover, adaptive time limits are imposed to avoid getting stuck on particularly hard second-stage problems, and a gap propagation between master problem and second-stage problems is used to stop solving them earlier if only a given non-zero optimality gap is to be reached overall. This makes our method particularly effective for problems where solving the second-stage problem is computationally challenging. To evaluate the method's performance, we compare it to two recent column-and-constraint generation methods from the literature on two applications: a robust capacitated location routing problem and a robust integrated berth allocation and quay crane assignment and scheduling problem. The first problem features a particularly hard second stage, and we show that our method is able to solve considerably more and larger instances in a given time limit. Using the second problem, we verify the general applicability of our method, even for problems where the second stage is relatively easy.

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arXiv2025

Patient Transport in Hospitals: A Literature Review of Operations Research and Management Science Methods

Most activities in hospitals require the presence of the patient. Delays in patient transport can disrupt operations, potentially resulting in idle staff, underutilized equipment, and postponed procedures, which in turn lead to lost revenue, unnecessary costs across many different areas and departments, and lower patient satisfaction. Consequently, patient transport planning is a central operational task in hospitals. This paper provides the first literature review of Operations Research and Management Science approaches for non-emergency, intra-hospital patient transport. We structure the different patient transport problems considered in the literature according to several main characteristics and introduce a five-field notation that allows for a concise representation of different problem variants. We then analyze the relevant literature with respect to different aspects related to the considered problem variant, the employed modeling and solution techniques, as well as the data used and the level of practical implementation achieved. Based on our literature analysis and semi-structured interviews with hospital practitioners, we compare current hospital practices and the existing literature, identify research gaps, and formulate an agenda for relevant future research.

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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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arXiv2024

Integrated Planning in Hospitals: A Review

Efficient planning of scarce resources in hospitals is a challenging task for which a large variety of Operations Research and Management Science approaches have been developed since the 1950s. While efficient planning of single resources such as operating rooms, beds, or specific types of staff can already lead to enormous efficiency gains, integrated planning of several resources has been shown to hold even greater potential, and a large number of integrated planning approaches have been presented in the literature over the past decades. This paper provides the first literature review that focuses specifically on the Operations Research and Management Science literature related to integrated planning of different resources in hospitals. We collect the relevant literature and analyze it regarding different aspects such as uncertainty modeling and the use of real-life data. Several cross comparisons reveal interesting insights concerning, e.g., relations between the modeling and solution methods used and the practical implementation of the approaches developed. Moreover, we provide a high-level taxonomy for classifying different resource-focused integration approaches and point out gaps in the literature as well as promising directions for future research.

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arXiv2024

Operational Research: Methods and Applications

Throughout its history, Operational Research has evolved to include a variety of methods, models and algorithms that have been applied to a diverse and wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first aims to summarise the up-to-date knowledge and provide an overview of the state-of-the-art methods and key developments in the various subdomains of the field. The second offers a wide-ranging list of areas where Operational Research has been applied. The article is meant to be read in a nonlinear fashion. It should be used as a point of reference or first-port-of-call for a diverse pool of readers: academics, researchers, students, and practitioners. The entries within the methods and applications sections are presented in alphabetical order. The authors dedicate this paper to the 2023 Turkey/Syria earthquake victims. We sincerely hope that advances in OR will play a role towards minimising the pain and suffering caused by this and future catastrophes.

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arXiv2023

Integrated patient-to-room and nurse-to-patient assignment in hospital wards

Assigning patients to rooms and nurses to patients are critical tasks within hospitals that directly affect patient and staff satisfaction, quality of care, and hospital efficiency. Both patient-to-room assignments and nurse-to-patient assignments are typically agreed upon at the ward level, and they interact in several ways such as jointly determining the walking distances nurses must cover between different patient rooms. This motivates to consider both problems jointly in an integrated fashion. This paper presents the first optimization models and algorithms for the integrated patient-to-room and nurse-to-patient assignment problem. We provide a mixed integer programming formulation of the integrated problem that considers the typical objectives from the single problems as well as additional objectives that can only be properly evaluated when integrating both problems. Moreover, motivated by the inherent complexity that results from integrating these two NP-hard and already computationally challenging problems, we devise an efficient heuristic for the integrated patient-to-room and nurse-to-patient assignment problem. To evaluate the running time and quality of the solution obtained with the heuristic, we conduct extensive computational experiments on both artificial and real-world instances. The artificial instances are generated by a parameterized instance generator for the integrated problem that is made freely available.

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arXiv2023

Efficiently Constructing Convex Approximation Sets in Multiobjective Optimization Problems

Convex approximation sets for multiobjective optimization problems are a well-studied relaxation of the common notion of approximation sets. Instead of approximating each image of a feasible solution by the image of some solution in the approximation set up to a multiplicative factor in each component, a convex approximation set only requires this multiplicative approximation to be achieved by some convex combination of finitely many images of solutions in the set. This makes convex approximation sets efficiently computable for a wide range of multiobjective problems - even for many problems for which (classic) approximations sets are hard to compute. In this article, we propose a polynomial-time algorithm to compute convex approximation sets that builds upon an exact or approximate algorithm for the weighted sum scalarization and is, therefore, applicable to a large variety of multiobjective optimization problems. The provided convex approximation quality is arbitrarily close to the approximation quality of the underlying algorithm for the weighted sum scalarization. In essence, our algorithm can be interpreted as an approximate variant of the dual variant of Benson's Outer Approximation Algorithm. Thus, in contrast to existing convex approximation algorithms from the literature, information on solutions obtained during the approximation process is utilized to significantly reduce both the practical running time and the cardinality of the returned solution sets while still guaranteeing the same worst-case approximation quality. We underpin these advantages by the first comparison of all existing convex approximation algorithms on several instances of the triobjective knapsack problem and the triobjective symmetric metric traveling salesman problem.

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arXiv2023

Approximating Multiobjective Optimization Problems: How exact can you be?

It is well known that, under very weak assumptions, multiobjective optimization problems admit $(1+\varepsilon,\dots,1+\varepsilon)$-approximation sets (also called $\varepsilon$-Pareto sets) of polynomial cardinality (in the size of the instance and in $\frac{1}{\varepsilon}$). While an approximation guarantee of $1+\varepsilon$ for any $\varepsilon>0$ is the best one can expect for singleobjective problems (apart from solving the problem to optimality), even better approximation guarantees than $(1+\varepsilon,\dots,1+\varepsilon)$ can be considered in the multiobjective case since the approximation might be exact in some of the objectives. Hence, in this paper, we consider partially exact approximation sets that require to approximate each feasible solution exactly, i.e., with an approximation guarantee of $1$, in some of the objectives while still obtaining a guarantee of $1+\varepsilon$ in all others. We characterize the types of polynomial-cardinality, partially exact approximation sets that are guaranteed to exist for general multiobjective optimization problems. Moreover, we study minimum-cardinality partially exact approximation sets concerning (weak) efficiency of the contained solutions and relate their cardinalities to the minimum cardinality of a $(1+\varepsilon,\dots,1+\varepsilon)$-approximation set.

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arXiv2023

Using Scalarizations for the Approximation of Multiobjective Optimization Problems: Towards a General Theory

We study the approximation of general multiobjective optimization problems with the help of scalarizations. Existing results state that multiobjective minimization problems can be approximated well by norm-based scalarizations. However, for multiobjective maximization problems, only impossibility results are known so far. Countering this, we show that all multiobjective optimization problems can, in principle, be approximated equally well by scalarizations. In this context, we introduce a transformation theory for scalarizations that establishes the following: Suppose there exists a scalarization that yields an approximation of a certain quality for arbitrary instances of multiobjective optimization problems with a given decomposition specifying which objective functions are to be minimized / maximized. Then, for each other decomposition, our transformation yields another scalarization that yields the same approximation quality for arbitrary instances of problems with this other decomposition. In this sense, the existing results about the approximation via scalarizations for minimization problems carry over to any other objective decomposition -- in particular, to maximization problems -- when suitably adapting the employed scalarization. We further provide necessary and sufficient conditions on a scalarization such that its optimal solutions achieve a constant approximation quality. We give an upper bound on the best achievable approximation quality that applies to general scalarizations and is tight for the majority of norm-based scalarizations applied in the context of multiobjective optimization. As a consequence, none of these norm-based scalarizations can induce approximation sets for optimization problems with maximization objectives, which unifies and generalizes the existing impossibility results concerning the approximation of maximization problems.

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arXiv2021

An Approximation Algorithm for a General Class of Multi-Parametric Optimization Problems

In a widely-studied class of multi-parametric optimization problems, the objective value of each solution is an affine function of real-valued parameters. Then, the goal is to provide an optimal solution set, i.e., a set containing an optimal solution for each non-parametric problem obtained by fixing a parameter vector. For many multi-parametric optimization problems, however, an optimal solution set of minimum cardinality can contain super-polynomially many solutions. Consequently, no polynomial-time exact algorithms can exist for these problems even if $\textsf{P}=\textsf{NP}$. We propose an approximation method that is applicable to a general class of multi-parametric optimization problems and outputs a set of solutions with cardinality polynomial in the instance size and the inverse of the approximation guarantee. This method lifts approximation algorithms for non-parametric optimization problems to their parametric version and provides an approximation guarantee that is arbitrarily close to the approximation guarantee of the approximation algorithm for the non-parametric problem. If the non-parametric problem can be solved exactly in polynomial time or if an FPTAS is available, our algorithm is an FPTAS. Further, we show that, for any given approximation guarantee, the minimum cardinality of an approximation set is, in general, not $\ell$-approximable for any natural number $\ell$ less or equal to the number of parameters, and we discuss applications of our results to classical multi-parametric combinatorial optimizations problems. In particular, we obtain an FPTAS for the multi-parametric minimum $s$-$t$-cut problem, an FPTAS for the multi-parametric knapsack problem, as well as an approximation algorithm for the multi-parametric maximization of independence systems problem.

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arXiv2021

Approximating Biobjective Minimization Problems Using General Ordering Cones

This article investigates the approximation quality achievable for biobjective minimization problems with respect to the Pareto cone by solutions that are (approximately) optimal with respect to larger ordering cones. When simultaneously considering $α$-approximations for all closed convex ordering cones of a fixed inner angle $γ\in [\frac π2, π]$, an approximation guarantee between $α$ and $2 α$ is achieved, which depends continuously on $γ$. The analysis is best-possible for any inner angle and it generalizes and unifies the known results that the set of supported solutions is a 2-approximation and that the efficient set itself is a 1-approximation. Moreover, it is shown that, for maximization problems, no approximation guarantee is achievable by considering larger ordering cones in the described fashion, which again generalizes a known result about the set of supported solutions.

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arXiv2021

The Product Knapsack Problem: Approximation and Complexity

We consider the product knapsack problem, which is the variant of the classical 0-1 knapsack problem where the objective consists of maximizing the product of the profits of the selected items. These profits are allowed to be positive or negative. We show that this recently introduced variant of the knapsack problem is weakly NP-hard and present a fully polynomial-time approximation scheme (FPTAS) for the problem. Moreover, we analyze the approximation quality achieved by a natural extension of the classical greedy procedure to the product knapsack problem.

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arXiv2020

The Power of the Weighted Sum Scalarization for Approximating Multiobjective Optimization Problems

We determine the power of the weighted sum scalarization with respect to the computation of approximations for general multiobjective minimization and maximization problems. Additionally, we introduce a new multi-factor notion of approximation that is specifically tailored to the multiobjective case and its inherent trade-offs between different objectives. For minimization problems, we provide an efficient algorithm that computes an approximation of a multiobjective problem by using an exact or approximate algorithm for its weighted sum scalarization. In case that an exact algorithm for the weighted sum scalarization is used, this algorithm comes arbitrarily close to the best approximation quality that is obtainable by supported solutions - both with respect to the common notion of approximation and with respect to the new multi-factor notion. Moreover, the algorithm yields the currently best approximation results for several well-known multiobjective minimization problems. For maximization problems, however, we show that a polynomial approximation guarantee can, in general, not be obtained in more than one of the objective functions simultaneously by supported solutions.

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