@misc{LenzSerrano, author = {Lenz, Ralf and Serrano, Felipe}, title = {Tight Convex Relaxations for the Expansion Planning Problem}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-81459}, abstract = {Secure energy transport is considered as highly relevant for the basic infrastructure of nowadays society and economy. To satisfy increasing demands and to handle more diverse transport situations, operators of energy networks regularly expand the capacity of their network by building new network elements, known as the expansion planning problem. A key constraint function in expansion planning problems is a nonlinear and nonconvex potential loss function. In order to improve the algorithmic performance of state-of-the-art MINLP solvers, this paper presents an algebraic description for the convex envelope of this function. Through a thorough computational study, we show that this tighter relaxation tremendously improve the performance of the MINLP solver SCIP on a large test set of practically relevant instances for the expansion planning problem. In particular, the results show that our achievements lead to an improvement of the solver performance for a development version by up to 58\%.}, language = {en} } @misc{Lenz, author = {Lenz, Ralf}, title = {Pipe Merging for Transient Gas Network Problems}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-82116}, abstract = {In practice, transient gas transport problems frequently have to be solved for large-scale gas networks. Gas network optimization problems typically belong to the class of Mixed-Integer Nonlinear Programming Problems (MINLP). However current state-of-the-art MINLP solvers are not yet mature enough to solve large-scale real-world instances. Therefore, an established approach in practice is to solve the problems with respect to a coarser representation of the network and thereby reducing the size of the underlying model. Two well-known aggregation methods that effectively reduce the network size are parallel and serial pipe merges. However, these methods have only been studied in stationary gas transport problems so far. This paper closes this gap and presents parallel and serial pipe merging methods in the context of transient gas transport. To this end, we introduce the concept of equivalent and heuristic subnetwork replacements. For the heuristic methods, we conduct a huge empirical evaluation based on real-world data taken from one of the largest gas networks in Europe. It turns out that both, parallel and serial pipe merging can be considered as appropriate aggregation methods for real-world transient gas flow problems.}, language = {en} } @misc{BorndoerferSchwartzSurau, author = {Bornd{\"o}rfer, Ralf and Schwartz, Stephan and Surau, William}, title = {Vertex Covering with Capacitated Trees}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-82616}, abstract = {The covering of a graph with (possibly disjoint) connected subgraphs is a fundamental problem in graph theory. In this paper, we study a version to cover a graph's vertices by connected subgraphs subject to lower and upper weight bounds, and propose a column generation approach to dynamically generate feasible and promising subgraphs. Our focus is on the solution of the pricing problem which turns out to be a variant of the NP-hard Maximum Weight Connected Subgraph Problem. We compare different formulations to handle connectivity, and find that a single-commodity flow formulation performs best. This is notable since the respective literature seems to have dismissed this formulation. We improve it to a new coarse-to-fine flow formulation that is theoretically and computationally superior, especially for large instances with many vertices of degree 2 like highway networks, where it provides a speed-up factor of 10 over the non-flow-based formulations. We also propose a preprocessing method that exploits a median property of weight constrained subgraphs, a primal heuristic, and a local search heuristic. In an extensive computational study we evaluate the presented connectivity formulations on different classes of instances, and demonstrate the effectiveness of the proposed enhancements. Their speed-ups essentially multiply to an overall factor of 20. Overall, our approach allows the reliabe solution of instances with several hundreds of nodes in a few minutes. These findings are further corroborated in a comparison to existing districting models on a set of test instances from the literature.}, language = {en} } @inproceedings{BorndoerferSchwartzSurau, author = {Bornd{\"o}rfer, Ralf and Schwartz, Stephan and Surau, William}, title = {Finding Minimum Balanced Separators - an Exact Approach}, series = {Operations Research Proceedings 2021}, booktitle = {Operations Research Proceedings 2021}, issn = {1438-0064}, doi = {https://doi.org/10.1007/978-3-031-08623-6_24}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-83138}, pages = {154 -- 159}, abstract = {Balanced separators are node sets that split the graph into size bounded components. They find applications in different theoretical and practical problems. In this paper we discuss how to find a minimum set of balanced separators in node weighted graphs. Our contribution is a new and exact algorithm that solves Minimum Balanced Separators by a sequence of Hitting Set problems. The only other exact method appears to be a mixed-integer program (MIP) for the edge weighted case. We adapt this model to node weighted graphs and compare it to our approach on a set of instances, resembling transit networks. It shows that our algorithm is far superior on almost all test instances.}, language = {en} } @misc{BorndoerferDenissenHelleretal., author = {Bornd{\"o}rfer, Ralf and Denißen, Jonas and Heller, Simon and Klug, Torsten and K{\"u}pper, Michael and Lindner, Niels and Reuther, Markus and Schlechte, Thomas and S{\"o}hlke, Andreas and Steadman, William}, title = {Microscopic Timetable Optimization for a Moving Block System}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-82547}, abstract = {We present an optimization model which is capable of routing and ordering trains on a microscopic level under a moving block regime. Based on a general timetabling definition (GTTP) that allows the plug in of arbitrarily detailed methods to compute running and headway times, we describe a layered graph approach using velocity expansion, and develop a mixed integer linear programming formulation. Finally, we present promising results for a German corridor scenario with mixed traffic, indicating that applying branch-and-cut to our model is able to solve reasonably sized instances with up to hundred trains to optimality.}, language = {en} } @misc{LindnerLiebchenMasing, author = {Lindner, Niels and Liebchen, Christian and Masing, Berenike}, title = {Forward Cycle Bases and Periodic Timetabling}, issn = {1438-0064}, doi = {10.4230/OASIcs.ATMOS.2021.2}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-82756}, abstract = {Periodic timetable optimization problems in public transport can be modeled as mixed-integer linear programs by means of the Periodic Event Scheduling Problem (PESP). In order to keep the branch-and-bound tree small, minimum integral cycle bases have been proven successful. We examine forward cycle bases, where no cycle is allowed to contain a backward arc. After reviewing the theory of these bases, we describe the construction of an integral forward cycle basis on a line-based event-activity network. Adding turnarounds to the instance \texttt{R1L1} of the benchmark library PESPlib, we computationally evaluate three types of forward cycle bases in the Pareto sense, and come up with significant improvements concerning dual bounds.}, language = {en} } @misc{LindnerMaristanydelasCasasSchiewe, author = {Lindner, Niels and Maristany de las Casas, Pedro and Schiewe, Philine}, title = {Optimal Forks: Preprocessing Single-Source Shortest Path Instances with Interval Data}, issn = {1438-0064}, doi = {10.4230/OASIcs.ATMOS.2021.7}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-82716}, abstract = {We investigate preprocessing for single-source shortest path queries in digraphs, where arc costs are only known to lie in an interval. More precisely, we want to decide for each arc whether it is part of some shortest path tree for some realization of costs. We show that this problem is solvable in polynomial time by giving a combinatorial algorithm, using optimal structures that we call forks. Our algorithm turns out to be very efficient in practice, and is sometimes even superior in quality to a heuristic developed for the one-to-one shortest path problem in the context of passenger routing in public transport.}, language = {en} } @misc{LindnervanLieshout, author = {Lindner, Niels and van Lieshout, Rolf}, title = {Benders Decomposition for the Periodic Event Scheduling Problem}, issn = {1438-0064}, doi = {10.1007/978-3-031-08623-6_43}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-83338}, abstract = {The Periodic Event Scheduling Problem (PESP) is the central mathematical model behind the optimization of periodic timetables in public transport. We apply Benders decomposition to the incidence-based MIP formulation of PESP. The resulting formulation exhibits particularly nice features: The subproblem is a minimum cost network flow problem, and feasibility cuts are equivalent to the well-known cycle inequalities by Odijk. We integrate the Benders approach into a branch-and-cut framework, and assess the performance of this method on instances derived from the benchmarking library PESPlib.}, language = {en} } @misc{FujiiItoKimetal., author = {Fujii, Koichi and Ito, Naoki and Kim, Sunyoung and Kojima, Masakazu and Shinano, Yuji and Toh, Kim-Chuan}, title = {Solving Challenging Large Scale QAPs}, issn = {1438-0064}, doi = {10.12752/8130}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-81303}, abstract = {We report our progress on the project for solving larger scale quadratic assignment problems (QAPs). Our main approach to solve large scale NP-hard combinatorial optimization problems such as QAPs is a parallel branch-and-bound method efficiently implemented on a powerful computer system using the Ubiquity Generator(UG) framework that can utilize more than 100,000 cores. Lower bounding procedures incorporated in the branch-and-bound method play a crucial role in solving the problems. For a strong lower bounding procedure, we employ the Lagrangian doubly nonnegative (DNN) relaxation and the Newton-bracketing method developed by the authors' group. In this report, we describe some basic tools used in the project including the lower bounding procedure and branching rules, and present some preliminary numerical results. Our next target problem is QAPs with dimension at least 50, as we have succeeded to solve tai30a and sko42 from QAPLIB for the first time.}, language = {en} } @misc{Bushe, type = {Master Thesis}, author = {Bushe, Julian}, title = {Rolling Stock Rotation Optimization with Maintenance Paths}, abstract = {Die Planung vom Zuguml{\"a}ufen ist eine der wichtigsten Aufgaben f{\"u}r Eisenbahnun- ternehmen. Dabei spielt auch die Einhaltung von vorgegebenen Wartungsintervallen eine zentrale Rolle f{\"u}r die Sicherheit und Zuverl{\"a}ssigkeit der Schienenfahrzeuge. Wir zeigen, wie man dieses Umlaufplanungsproblem unter Beachtung von Wartungsbe- dingungen mathematisch formuliert, modelliert und l{\"o}st — sowohl in der Theorie als auch im Anwendungsfall mit Szenarien der DB Fernverkehr AG, einer Konzern- tochter der Deutschen Bahn f{\"u}r den Schienenpersonenfernverkehr. Markus Reuther hat sich in seiner Dissertation [11] mit diesem Problem besch{\"a}ftigt und es mit Hilfe eines passenden Hypergraphen als gemischt-ganzzahliges Programm modelliert. Neben der Modellierung pr{\"a}sentiert Reuther in seiner Arbeit neuartige algorithmische Ideen, darunter den sogenannten Coarse-to-Fine -Ansatz, bei dem zun{\"a}chst Teile des Problems auf einer weniger detaillierten ( coarse ) Ebene gel{\"o}st werden und diese L{\"o}sung dann verwendet wird, um auf effiziente Art und Weise eine L{\"o}sung f{\"u}r das urspr{\"u}ngliche Problem zu finden. Zur Wartungsplanung nutzt Reuther einen Fluss im Hypergraphen, der den Ressourcenverbrauch der Fahrzeuge modelliert. In der linearen Relaxierung des Modells f{\"u}hrt dies dazu, dass die Zahl der notwendigen Wartungen systematisch untersch{\"a}tzt wird. Dadurch bleibt in vielen F{\"a}llen eine große L{\"u}cke zwischen dem Zielfunktionswert einer optimalen L{\"o}sung des ganzzahligen Problems und der untere Schranke, die uns die lineare Relaxierung liefert. Wir nehmen uns in dieser Arbeit dieses Problems an. Wir entwickeln ein auf Pfaden basierendes ganzzahliges Modell f{\"u}r das Umlaufplanungsproblem und zeigen, dass die untere Schranke mindestens so scharf oder sch{\"a}rfer ist als die untere Schranke, die das Modell von Reuther liefert. Um das Modell zu l{\"o}sen, entwickeln wir einen Algorithmus, der Spaltengenerierung mit dem Coarse-to-Fine-Ansatz von Reuther verbindet. Weiterhin entwickeln wir eine Spaltenauswahlregel zur Beschleunigung des Algorithmus. Das Modell und alle in der Arbeit vorgestellten Algorithmen wur- den im Rahmen der Arbeit implementiert und mit Anwendungsszenarien der DB Fernverkehr AG getestet. Unsere Tests zeigen, dass unser Modell f{\"u}r fast alle Szena- rien deutlich sch{\"a}rfere untere Schranken liefert als das Modell von Reuther. In den getesteten Instanzen konnten wir durch die Verbesserung der unteren Schranke bis zu 99\% der Optimalit{\"a}tsl{\"u}cke schließen. In einem Drittel der F{\"a}lle konnten wir durch unseren Ansatz auch f{\"u}r das ganzzahlige Programm verbesserte Zielfunktionswerte erreichen}, language = {en} } @masterthesis{Lange, type = {Bachelor Thesis}, author = {Lange, Johanna}, title = {A Decomposition and Dualization Approach to the Periodic Event Scheduling Problem}, abstract = {Scheduling ist ein wichtiger Forschungsgegenstand im Bereich der diskreten Optimierung. Es geht darum, einen Schedule, d.h. einen Ablaufplan, f{\"u}r gegebene Ereignisse zu finden. Dieser soll optimal hinsichtlich einer Zielfunktion wie zum Beispiel minimaler Dauer oder Kosten sein. Dabei gibt es in der Regel Nebenbedingungen wie Vorrangbeziehungen zwischen den Ereignissen oder zeitliche Einschr{\"a}nkungen, die zu erf{\"u}llen sind. Falls die Ereignisse periodisch wiederkehren, spricht man von periodischem Scheduling. Beispiele sind das Erstellen von Zugfahrpl{\"a}nen, die Schaltungvon Ampelsignalen oder die Planung von Produktionsabl{\"a}ufen. Mathematisch k{\"o}nnen diese Probleme mit dem Periodic Event Scheduling Problem (PESP) modelliert werden, das als gemischt-ganzzahliges Programm formuliert werden kann. In dieser Bachelorarbeit wird ein Ansatz zur L{\"o}sung des PESP mittels Zerlegung und Dualisierung entwickelt. In den Kapiteln 2 und 3 werden zun{\"a}chst die notwendigen graphentheoretischen Grundlagen und das PESP eingef{\"u}hrt. In Kapitel 4 wird das PESP durch Fixierung der ganzzahligen Variablen in lineare Programme zerlegt. Dieses Unterproblem wird dualisiert und wieder in das PESP eingesetzt. Daf{\"u}r ist eine weitere Nebenbedingung n{\"o}tig. Im f{\"u}nften Kapitel behandeln wir die L{\"o}sung des teildualisierten PESP. Eine M{\"o}glichkeit ist es, sich auf eine Teilmenge der Nebenbedingungen zu beschr{\"a}nken. Eine weitere M{\"o}glichkeit ist ein Algorithmus, der{\"a}hnlich wie BendersZerlegung die Nebenbedingungen dynamisch erzeugt. Dieser Algorithmus wird in Kapitel 6 implementiert und an vier Beispielen getestet.}, language = {en} } @misc{Kuehner, type = {Master Thesis}, author = {K{\"u}hner, Arno}, title = {Shortest Paths with Boolean Constraints}, abstract = {For this thesis we study the Constrained Horizontal Flightplanning Problem (CHFPP) for which one has to find the path of minimum cost between airports s and t in a directed graph that respects a set of boolean constraints. To this end we give a survey of three different multilabel algorithms that all use a domination subroutine. We summarize an approach by Knudsen, Chiarandini and Larsen to define this domination and afterwards present our own method which builds on that approach. We suggest different implementation techniques to speed up the computation time, most notably a Reoptimization for an iterative method to solve the problem. Furthermore we implemented the different versions of the algorithm and present statistics on their computation as well as an overview of statistics on the set of real-world constraints that we were given. Finally we present two alternative approaches that tackle the problem, a heuristic with similarities to a Lagrangian relaxation and an approach that makes use of an algorithm which finds the k shortest path of a graph such as the ones of Epstein or Yen.}, language = {en} } @misc{Bortoletto, type = {Master Thesis}, author = {Bortoletto, Enrico}, title = {The tropical tiling of periodic timetable space and a dual modulo network simplex algorithm}, abstract = {We propose a tropical interpretation of the solution space of the Periodic Event Scheduling Problem as a collection of polytropes, making use of the characterization of tropical cones as weighted digraph polyhedra. General and geometric properties of the polytropal collection are inspected and understood in connection with the combinatorial properties of the underlying periodic event scheduling instance. Novel algorithmic ideas are presented and tested, making use of the aforementioned theoretical results to solve and optimize the problem.}, language = {en} } @masterthesis{Kraus, type = {Bachelor Thesis}, author = {Kraus, Luitgard}, title = {A Label Setting Multiobjective Shortest Path FPTAS}, abstract = {Algorithms that solve the shortest path problem can largely be split into the two categories of label setting and label correcting. The Multiobjective Shortest Path (MOSP) problem is a generalization of the classical shortest path problem in terms of the dimension of the cost function. We explore the differences of two similar MOSP label setting algorithms. Furthermore, we present and prove a general method of how to derive Fully Polynomial Time Approximation Schemes (FPTAS) for MOSP label setting algorithms. Finally, we explore two pruning techniques for the one to one variants of exact label setting MOSP algorithms and adapt them to their FPTAS variants.}, language = {en} } @misc{Rahmati, type = {Master Thesis}, author = {Rahmati, Niloofar}, title = {Resource Constrained APSP-Algorithm with Possible Reloading Stops}, language = {en} } @misc{BorndoerferSchwartzSurau, author = {Bornd{\"o}rfer, Ralf and Schwartz, Stephan and Surau, William}, title = {Rooted Maximum Weight Connected Subgraphs with Balancing and Capacity Constraints}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-84427}, language = {en} } @misc{LindnerLiebchen, author = {Lindner, Niels and Liebchen, Christian}, title = {Timetable Merging for the Periodic Event Scheduling Problem}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-81587}, abstract = {We propose a new mixed integer programming based heuristic for computing new benchmark primal solutions for instances of the PESPlib. The PESPlib is a collection of instances for the Periodic Event Scheduling Problem (PESP), comprising periodic timetabling problems inspired by real-world railway timetabling settings, and attracting several international research teams during the last years. We describe two strategies to merge a set of good periodic timetables. These make use of the instance structure and minimum weight cycle bases, finally leading to restricted mixed integer programming formulations with tighter variable bounds. Implementing this timetable merging approach in a concurrent solver, we improve the objective values of the best known solutions for the smallest and largest PESPlib instances by 1.7 and 4.3 percent, respectively.}, language = {en} } @misc{EiflerGleixnerPulaj, author = {Eifler, Leon and Gleixner, Ambros and Pulaj, Jonad}, title = {A Safe Computational Framework for Integer Programming applied to Chv{\´a}tal's Conjecture}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-84444}, abstract = {We describe a general and safe computational framework that provides integer programming results with the degree of certainty that is required for machine-assisted proofs of mathematical theorems. At its core, the framework relies on a rational branch-and-bound certificate produced by an exact integer programming solver, SCIP, in order to circumvent floating-point roundoff errors present in most state-of-the-art solvers for mixed-integer programs. The resulting certificates are self-contained and checker software exists that can verify their correctness independently of the integer programming solver used to produce the certificate. This acts as a safeguard against programming errors that may be present in complex solver software. The viability of this approach is tested by applying it to finite cases of Chv{\´a}tal's conjecture, a long-standing open question in extremal combinatorics. We take particular care to verify also the correctness of the input for this specific problem, using the Coq formal proof assistant. As a result we are able to provide a first machine-assisted proof that Chv{\´a}tal's conjecture holds for all downsets whose union of sets contains seven elements or less.}, language = {en} } @misc{EulerBorndoerferStrunketal., author = {Euler, Ricardo and Bornd{\"o}rfer, Ralf and Strunk, Timo and Takkula, Tuomo}, title = {ULD Build-Up Scheduling with Dynamic Batching in an Air Freight Hub}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-83482}, abstract = {Air freight is usually shipped in standardized unit load devices (ULDs). The planning process for the consolidation of transit cargo from inbound flights or locally emerging shipments into ULDs for outbound flights is called build-up scheduling. More specifically, outbound ULDs must be assigned a time and a workstation subject to both workstation capacity constraints and the availability of shipments which in turn depends on break-down decisions for incoming ULDs. ULDs scheduled for the same outbound flight should be built up in temporal and spatial proximity. This serves both to minimize overhead in transportation times and to allow workers to move freight between ULDs. We propose to address this requirement by processing ULDs for the same outbound flight in batches. For the above build-up scheduling problem, we introduce a multi-commodity network design model. Outbound flights are modeled as commodities; transit cargo is represented by cargo flow volume and unpack and batch decisions are represented as design variables. The model is solved with standard MIP solvers on a set of benchmark data. For instances with a limited number of resource conflicts, near-optimal solutions are found in under two hours for a whole week of operations.}, language = {en} } @misc{Francobaldi, type = {Master Thesis}, author = {Francobaldi, Matteo}, title = {Learning to Use Local Cuts}, abstract = {We propose a machine learning approach to address a specific algorithmic question that arises during the solving process of a mixed-integer linear programming problem, namely, whether to use cutting planes only at the root node or also at internal nodes of the branch-and-bound search tree, or equivalently, whether to run a cut-and-branch or rather a branch-and-cut algorithm. Within a supervised regression framework, we develop three machine learning models, Linear Model, Random Forest and Neural Network, for predicting the relative performance between the two methods, local-cut and no-local-cut. Hence, through an extensive computational study conducted with FICO Xpress over a large test bed of problems, we evaluate the produced strategies, and we show that they are able to provide, upon the existing policies, a significant improvement to the performance of the solver. In fact, a variant of the random forest suggested in the present work has already been implemented by the development team of Xpress, and released with version 8.13 of the software.}, language = {en} } @misc{EiflerGleixner, author = {Eifler, Leon and Gleixner, Ambros}, title = {A Computational Status Update for Exact Rational Mixed Integer Programming}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-81298}, abstract = {The last milestone achievement for the roundoff-error-free solution of general mixed integer programs over the rational numbers was a hybrid-precision branch-and-bound algorithm published by Cook, Koch, Steffy, and Wolter in 2013. We describe a substantial revision and extension of this framework that integrates symbolic presolving, features an exact repair step for solutions from primal heuristics, employs a faster rational LP solver based on LP iterative refinement, and is able to produce independently verifiable certificates of optimality. We study the significantly improved performance and give insights into the computational behavior of the new algorithmic components. On the MIPLIB 2017 benchmark set, we observe an average speedup of 6.6x over the original framework and 2.8 times as many instances solved within a time limit of two hours.}, language = {en} }