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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.

This paper presents a new scenario addition method for two-stage robust mixed-integer programs with finite uncertainty sets. Our method combines and extends speed-up techniques used in previous scenario addition methods (also called 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 scenario addition 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.

Objective - This work introduces Dareplane, a modular and broad technology-agnostic open source software platform for brain-computer interface research with an application focus on adaptive deep brain stimulation (aDBS). One difficulty for investigating control approaches for aDBS resides with the complex setups required for aDBS experiments, a challenge Dareplane tries to address. Approach - The key features of the platform are presented and the composition of modules into a full experimental setup is discussed in the context of a Python-based orchestration module. The performance of a typical experimental setup on Dareplane for aDBS is evaluated in three benchtop experiments, covering (a) an easy-to-replicate setup using an Arduino microcontroller, (b) a setup with hardware of an implantable pulse generator, and (c) a setup using an established and CE certified external neurostimulator. The full technical feasibility of the platform in the aDBS context is demonstrated in a first closed-loop session with externalized leads on a patient with Parkinson's disease receiving DBS treatment and further in a non-invasive BCI speller application using code-modulated visual evoked responses (c-VEP). Main results - The platform is implemented and open-source accessible on https://github.com/bsdlab/Dareplane. Benchtop results show that performance of the platform is sufficient for current aDBS latencies, and the platform could successfully be used in the aDBS experiment. The timing-critical c-VEP speller could be successfully implemented on the platform achieving expected information transfer rates. Significance - The Dareplane platform supports aDBS setups, and more generally the research on neurotechnological systems such as brain-computer interfaces. It provides a modular, technology-agnostic, and easy-to-implement software platform to make experimental setups more resilient and replicable.

As brain-computer interfacing (BCI) systems transition from assistive technology to more diverse applications, their speed, reliability, and user experience become increasingly important. Dynamic stopping methods enhance BCI system speed by deciding at any moment whether to output a result or wait for more information. Such approach leverages trial variance, allowing good trials to be detected earlier, thereby speeding up the process without significantly compromising accuracy. Existing dynamic stopping algorithms typically optimize measures such as symbols per minute (SPM) and information transfer rate (ITR). However, these metrics may not accurately reflect system performance for specific applications or user types. Moreover, many methods depend on arbitrary thresholds or parameters that require extensive training data. We propose a model-based approach that takes advantage of the analytical knowledge that we have about the underlying classification model. By using a risk minimisation approach, our model allows precise control over the types of errors and the balance between precision and speed. This adaptability makes it ideal for customizing BCI systems to meet the diverse needs of various applications. We validate our proposed method on a publicly available dataset, comparing it with established static and dynamic stopping methods. Our results demonstrate that our approach offers a broad range of accuracy-speed trade-offs and achieves higher precision than baseline stopping 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.

A limitation of brain-computer interface (BCI) spellers is that they require the user to be able to move the eyes to fixate on targets. This poses an issue for users who cannot voluntarily control their eye movements, for instance, people living with late-stage amyotrophic lateral sclerosis (ALS). This pilot study makes the first step towards a gaze-independent speller based on the code-modulated visual evoked potential (c-VEP). Participants were presented with two bi-laterally located stimuli, one of which was flashing, and were tasked to attend to one of these stimuli either by directly looking at the stimuli (overt condition) or by using spatial attention, eliminating the need for eye movement (covert condition). The attended stimuli were decoded from electroencephalography (EEG) and classification accuracies of 88% and 100% were obtained for the covert and overt conditions, respectively. These fundamental insights show the promising feasibility of utilizing the c-VEP protocol for gaze-independent BCIs that use covert spatial attention when both stimuli flash simultaneously., Comment: 6 pages, 3 figures, 9th Graz Brain-Computer Interface Conference 2024

Auditory attention decoding (AAD) aims to extract from brain activity the attended speaker amidst candidate speakers, offering promising applications for neuro-steered hearing devices and brain-computer interfacing. This pilot study makes a first step towards AAD using the noise-tagging stimulus protocol, which evokes reliable code-modulated evoked potentials, but is minimally explored in the auditory modality. Participants were sequentially presented with two Dutch speech stimuli that were amplitude-modulated with a unique binary pseudo-random noise-code, effectively tagging these with additional decodable information. We compared the decoding of unmodulated audio against audio modulated with various modulation depths, and a conventional AAD method against a standard method to decode noise-codes. Our pilot study revealed higher performances for the conventional method with 70 to 100 percent modulation depths compared to unmodulated audio. The noise-code decoder did not further improve these results. These fundamental insights highlight the potential of integrating noise-codes in speech to enhance auditory speaker detection when multiple speakers are presented simultaneously., Comment: 6 pages, 2 figures, 9th Graz Brain-Computer Interface Conference 2024

This study explores two zero-training methods aimed at enhancing the usability of brain-computer interfaces (BCIs) by eliminating the need for a calibration session. We introduce a novel method rooted in the event-related potential (ERP) domain, unsupervised mean maximization (UMM), to the fast code-modulated visual evoked potential (c-VEP) stimulus protocol. We compare UMM to the state-of-the-art c-VEP zero-training method that uses canonical correlation analysis (CCA). The comparison includes instantaneous classification and classification with cumulative learning from previously classified trials for both CCA and UMM. Our study shows the effectiveness of both methods in navigating the complexities of a c-VEP dataset, highlighting their differences and distinct strengths. This research not only provides insights into the practical implementation of calibration-free BCI methods but also paves the way for further exploration and refinement. Ultimately, the fusion of CCA and UMM holds promise for enhancing the accessibility and usability of BCI systems across various application domains and a multitude of stimulus protocols., Comment: 6 pages, 2 figures, 9th Graz Brain-Computer Interface Conference 2024

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.

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.

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.

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.

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.

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.

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.

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 $\alpha$-approximations for all closed convex ordering cones of a fixed inner angle $\gamma \in [\frac \pi 2, \pi]$, an approximation guarantee between $\alpha$ and $2 \alpha$ is achieved, which depends continuously on $\gamma$. 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.

In recent years, the congruence $$ \sum_{\substack{i+j+k=p\\ i,j,k>0}} \frac1{ijk} \equiv -2 B_{p-3} \pmod{p}, $$ first discovered by the last author have been generalized by either increasing the number of indices and considering the corresponding supercongruences, or by considering the alternating version of multiple harmonic sums. In this paper, we prove a family of similar supercongruences modulo prime powers $p^r$ with the indexes summing up to $mp^r$ where $m$ is coprime to $p$, where all the indexes are also coprime to $p$., Comment: 16 pages, final version for publication

Papadimitriou and Yannakakis show that the polynomial-time solvability of a certain singleobjective problem determines the class of multiobjective optimization problems that admit a polynomial-time computable $(1+\varepsilon, \dots , 1+\varepsilon)$-approximate Pareto set (also called an $\varepsilon$-Pareto set). Similarly, in this article, we characterize the class of problems having a polynomial-time computable approximate $\varepsilon$-Pareto set that is exact in one objective by the efficient solvability of an appropriate singleobjective problem. This class includes important problems such as multiobjective shortest path and spanning tree, and the approximation guarantee we provide is, in general, best possible. Furthermore, for biobjective problems from this class, we provide an algorithm that computes a one-exact $\varepsilon$-Pareto set of cardinality at most twice the cardinality of a smallest such set and show that this factor of 2 is best possible. For three or more objective functions, however, we prove that no constant-factor approximation on the size of the set can be obtained efficiently.

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.

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.

We analyze the computational complexity of the many types of pencil-and-paper-style puzzles featured in the 2016 puzzle video game The Witness. In all puzzles, the goal is to draw a simple path in a rectangular grid graph from a start vertex to a destination vertex. The different puzzle types place different constraints on the path: preventing some edges from being visited (broken edges); forcing some edges or vertices to be visited (hexagons); forcing some cells to have certain numbers of incident path edges (triangles); or forcing the regions formed by the path to be partially monochromatic (squares), have exactly two special cells (stars), or be singly covered by given shapes (polyominoes) and/or negatively counting shapes (antipolyominoes). We show that any one of these clue types (except the first) is enough to make path finding NP-complete ("witnesses exist but are hard to find"), even for rectangular boards. Furthermore, we show that a final clue type (antibody), which necessarily "cancels" the effect of another clue in the same region, makes path finding $\Sigma_2$-complete ("witnesses do not exist"), even with a single antibody (combined with many anti/polyominoes), and the problem gets no harder with many antibodies. On the positive side, we give a polynomial-time algorithm for monomino clues, by reducing to hexagon clues on the boundary of the puzzle, even in the presence of broken edges, and solving "subset Hamiltonian path" for terminals on the boundary of an embedded planar graph in polynomial time., Comment: 72 pages, 59 figures. Revised proof of Lemma 3.5. A short version of this paper appeared at the 9th International Conference on Fun with Algorithms (FUN 2018)

We present a general technique for approximating bicriteria minimization problems with positive-valued, polynomially computable objective functions. Given $0<\epsilon\leq1$ and a polynomial-time $\alpha$-approximation algorithm for the corresponding weighted sum problem, we show how to obtain a bicriteria $(\alpha\cdot(1+2\epsilon),\alpha\cdot(1+\frac{2}{\epsilon}))$-approximation algorithm for the budget-constrained problem whose running time is polynomial in the encoding length of the input and linear in $\frac{1}{\epsilon}$. Moreover, we show that our method can be extended to compute an $(\alpha\cdot(1+2\epsilon),\alpha\cdot(1+\frac{2}{\epsilon}))$-approximate Pareto curve under the same assumptions. Our technique applies to many minimization problems to which most previous algorithms for computing approximate Pareto curves cannot be applied because the corresponding gap problem is $\textsf{NP}$-hard to solve. For maximization problems, however, we show that approximation results similar to the ones presented here for minimization problems are impossible to obtain in polynomial time unless $\textsf{P}=\textsf{NP}$.

We investigate the complexity and approximability of the budget-constrained minimum cost flow problem, which is an extension of the traditional minimum cost flow problem by a second kind of costs associated with each edge, whose total value in a feasible flow is constrained by a given budget B. This problem can, e.g., be seen as the application of the {\epsilon}-constraint method to the bicriteria minimum cost flow problem. We show that we can solve the problem exactly in weakly polynomial time $O(\log M \cdot MCF(m,n,C,U))$, where C, U, and M are upper bounds on the largest absolute cost, largest capacity, and largest absolute value of any number occuring in the input, respectively, and MCF(m,n,C,U) denotes the complexity of finding a traditional minimum cost flow. Moreover, we present two fully polynomial-time approximation schemes for the problem on general graphs and one with an improved running-time for the problem on acyclic graphs.

We present a specialized network simplex algorithm for the budget-constrained minimum cost flow problem, which is an extension of the traditional minimum cost flow problem by a second kind of costs associated with each edge, whose total value in a feasible flow is constrained by a given budget B. We present a fully combinatorial description of the algorithm that is based on a novel incorporation of two kinds of integral node potentials and three kinds of reduced costs. We prove optimality criteria and combine two methods that are commonly used to avoid cycling in traditional network simplex algorithms into new techniques that are applicable to our problem. With these techniques and our definition of the reduced costs, we are able to prove a pseudo-polynomial running time of the overall procedure, which can be further improved by incorporating Dantzig's pivoting rule. Moreover, we present computational results that compare our procedure with Gurobi.

We developed task-optimized deep neural networks (DNNs) that achieved state-of-the-art performance in different evaluation scenarios for automatic music tagging. These DNNs were subsequently used to probe the neural representations of music. Representational similarity analysis revealed the existence of a representational gradient across the superior temporal gyrus (STG). Anterior STG was shown to be more sensitive to low-level stimulus features encoded in shallow DNN layers whereas posterior STG was shown to be more sensitive to high-level stimulus features encoded in deep DNN layers.

The excitation of surface plasmons in magnetic nano-structures can strongly influence their magneto-optical properties. Here, we use photoemission electron microscopy to map the spatial distribution of the electric near-field on a nano-patterned magnetic surface that supports plasmon polaritons. By using different photon energies and polarization states of the incident light we reveal that the electric near-field is either concentrated in spots forming a hexagonal lattice with the same symmetry as the Ni nano-pattern or in stripes oriented along the $\Gamma$-K direction of the lattice and perpendicular to the polarization direction. We show that the polarization-dependent near-field enhancement on the patterned surface is directly correlated to both the excitation of surface plasmon polaritons on the patterned surface as well as the enhancement of the polar magneto-optical Kerr effect.