arXiv subjects

Stephan Helfrich

Publications and source records attributed to Stephan Helfrich.

5 recordsLinked to original sources

arXiv2026

A Defender-Attacker-Defender Model for Optimizing the Resilience of Hospital Networks to Cyberattacks

Considering the increasing frequency of cyberattacks affecting multiple hospitals simultaneously, improving resilience at a network level is essential. Various countermeasures exist to improve resilience against cyberattacks, such as deploying controls that strengthen IT infrastructures to limit their impact, or enabling resource sharing, patient transfers and backup capacities to maintain services of hospitals in response to realized attacks. However, determining the most cost-effective combination among these wide range of countermeasures is a complex challenge, further intensified by constrained budgets and competing priorities between maintaining efficient daily hospital operations and investing in disaster preparedness. To address these challenges, we propose a defender-attacker-defender optimization model that supports decision-makers in identifying effective strategies for improving the resilience of a network of hospitals against cyberattacks. The model explicitly captures interdependence between hospital services and their supporting IT infrastructures. By doing so, cyberattacks can be directly translated into reductions of service capacities, which allows to assess proactive and reactive strategies on both the operational and technical sides within a single framework. Further, time-dependent resilience measures are incorporated as design objectives to account for the mid- to long-term consequences of cyberattacks. The model is validated based on the German hospital network, suggesting that enabling cooperation with backup capacities particularly in urban areas, alongside strengthening of IT infrastructures across all hospitals, are crucial strategies.

cs.CR↗

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.

math.OC↗

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.

math.OC↗

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 $\alpha$-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.

math.OC↗

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.

math.OC↗