Robust Reoptimization of Steiner Trees

In reoptimization problems, one is given an optimal solution to a problem instance and a local modification of the instance. The goal is to obtain a solution for the modified instance. The additional information about the instance provided by the given solution plays a central role: we aim to use that information in order to obtain better solutions than we are able to compute from scratch. In this paper, we consider Steiner tree reoptimization and address the optimality requirement of the provided solution. Instead of assuming that we are provided an optimal solution, we relax the assumption to the more realistic scenario where we are given an approximate solution with an upper bound on its performance guarantee. We show that for Steiner tree reoptimization there is a clear separation between local modifications where optimality is crucial for obtaining improved approximations and those instances where approximate solutions are acceptable starting points. For some of the local modifications that have been considered in previous research, we show that for every fixed epsilon > 0, approximating the reoptimization problem with respect to a given (1+epsilon)-approximation is as hard as approximating the Steiner tree problem itself (whereas with a given optimal solution to the original problem it is known that one can obtain considerably improved results). Furthermore, we provide a new algorithmic technique that, with some further insights, allows us to obtain improved performance guarantees for Steiner tree reoptimization with respect to all remaining local modifications that have been considered in the literature: a required node of degree more than one becomes a Steiner node; a Steiner node becomes a required node; the cost of one edge is increased

Robust Reoptimization of Steiner Trees

In reoptimization, one is given an optimal solution to a problem instance and a (locally) modified instance. The goal is to obtain a solution for the modified instance. We aim to use information obtained from the given solution in order to obtain a better solution for the new instance than we are able to compute from scratch. In this paper, we consider Steiner tree reoptimization and address the optimality requirement of the provided solution. Instead of assuming that we are provided an optimal solution, we relax the assumption to the more realistic scenario where we are given an approximate solution with an upper bound on its performance guarantee. We show that for Steiner tree reoptimization there is a clear separation between local modifications where optimality is crucial for obtaining improved approximations and those instances where approximate solutions are acceptable starting points. For some of the local modifications that have been considered in previous research, we show that for every fixed ε>0, approximating the reoptimization problem with respect to a given (1+ε)-approximation is as hard as approximating the Steiner tree problem itself. In contrast, with a given optimal solution to the original problem it is known that one can obtain considerably improved results. Furthermore, we provide a new algorithmic technique that, with some further insights, allows us to obtain improved performance guarantees for Steiner tree reoptimization with respect to all remaining local modifications that have been considered in the literature: a required node of degree more than one becomes a Steiner node; a Steiner node becomes a required node; the cost of one edge is increased

New algorithms for Steiner tree reoptimization

Reoptimization is a setting in which we are given an (near) optimal solution of a problem instance and a local modification that slightly changes the instance. The main goal is that of finding an (near) optimal solution of the modified instance. We investigate one of the most studied scenarios in reoptimization known as Steiner tree reoptimization. Steiner tree reoptimization is a collection of strongly NP-hard optimization problems that are defined on top of the classical Steiner tree problem and for which several constant-factor approximation algorithms have been designed in the last decade. In this paper we improve upon all these results by developing a novel technique that allows us to design polynomial-time approximation schemes. Remarkably, prior to this paper, no approximation algorithm better than recomputing a solution from scratch was known for the elusive scenario in which the cost of a single edge decreases. Our results are best possible since none of the problems addressed in this paper admits a fully polynomial-time approximation scheme, unless P=NP

Arc routing problems: A review of the past, present, and future

[EN] Arc routing problems (ARPs) are defined and introduced. Following a brief history of developments in this area of research, different types of ARPs are described that are currently relevant for study. In addition, particular features of ARPs that are important from a theoretical or practical point of view are discussed. A section on applications describes some of the changes that have occurred from early applications of ARP models to the present day and points the way to emerging topics for study. A final section provides information on libraries and instance repositories for ARPs. The review concludes with some perspectives on future research developments and opportunities for emerging applicationsThis research was supported by the Ministerio de Economia y Competitividad and Fondo Europeo de Desarrollo Regional, Grant/Award Number: PGC2018-099428-B-I00. The Research Council of Norway, Grant/Award Numbers: 246825/O70 (DynamITe), 263031/O70 (AXIOM).Corberán, Á.; Eglese, R.; Hasle, G.; Plana, I.; Sanchís Llopis, JM. (2021). Arc routing problems: A review of the past, present, and future. Networks. 77(1):88-115. https://doi.org/10.1002/net.21965S8811577

Mathematical Methods and Operation Research in Logistics, Project Planning, and Scheduling

In the last decade, the Industrial Revolution 4.0 brought flexible supply chains and flexible design projects to the forefront. Nevertheless, the recent pandemic, the accompanying economic problems, and the resulting supply problems have further increased the role of logistics and supply chains. Therefore, planning and scheduling procedures that can respond flexibly to changed circumstances have become more valuable both in logistics and projects. There are already several competing criteria of project and logistic process planning and scheduling that need to be reconciled. At the same time, the COVID-19 pandemic has shown that even more emphasis needs to be placed on taking potential risks into account. Flexibility and resilience are emphasized in all decision-making processes, including the scheduling of logistic processes, activities, and projects