@article{PedersenSpreckelsenGotzesetal.2023, author = {Pedersen, Jaap and Spreckelsen, Klaus and Gotzes, Uwe and Zittel, Janina and Koch, Thorsten}, title = {Beimischung von Wasserstoff zum Erdgas: Eine Kapazit{\"a}tsstudie des deutschen Gasnetzes}, journal = {gwf Gas + Energie}, edition = {06/2023}, publisher = {Vulkan Verlag}, year = {2023}, abstract = {Die europaische Gasinfrastruktur wird disruptiv in ein zukunftiges dekarbonisiertes Energiesystem ver{\"a}ndert; ein Prozess, der angesichts der j{\"u}ngsten politischen Situation beschleunigt werden muss. Mit einem wachsenden Wasserstoffmarkt wird der pipelinebasierte Transport unter Nutzung der bestehenden Erdgasinfrastruktur wirtschaftlich sinnvoll, tr{\"a}gt zur Erh{\"o}hung der {\"o}ffentlichen Akzeptanz bei und beschleunigt den Umstellungsprozess. In diesem Beitrag wird die maximal technisch machbare Einspeisung von Wasserstoff in das bestehende deutsche Erdgastransportnetz hinsichtlich regulatorischer Grenzwerte der Gasqualit{\"a}t analysiert. Die Analyse erfolgt auf Basis eines transienten Tracking-Modells, das auf dem allgemeinen Pooling-Problem einschließlich Linepack aufbaut. Es zeigt sich, dass das Gasnetz auch bei strengen Grenzwerten gen ̈ugend Kapazit{\"a}t bietet, um f{\"u}r einen großen Teil der bis 2030 geplanten Erzeugungskapazit{\"a}t f{\"u}r gr{\"u}nen Wasserstoff als garantierter Abnehmer zu dienen.}, language = {de} } @article{RehfeldtKochShinano2023, author = {Rehfeldt, Daniel and Koch, Thorsten and Shinano, Yuji}, title = {Faster exact solution of sparse MaxCut and QUBO problems}, volume = {15}, journal = {Mathematical Programming Computation}, doi = {10.1007/s12532-023-00236-6}, pages = {445 -- 470}, year = {2023}, abstract = {The maximum-cut problem is one of the fundamental problems in combinatorial optimization. With the advent of quantum computers, both the maximum-cut and the equivalent quadratic unconstrained binary optimization problem have experienced much interest in recent years. This article aims to advance the state of the art in the exact solution of both problems—by using mathematical programming techniques. The main focus lies on sparse problem instances, although also dense ones can be solved. We enhance several algorithmic components such as reduction techniques and cutting-plane separation algorithms, and combine them in an exact branch-and-cut solver. Furthermore, we provide a parallel implementation. The new solver is shown to significantly outperform existing state-of-the-art software for sparse maximum-cut and quadratic unconstrained binary optimization instances. Furthermore, we improve the best known bounds for several instances from the 7th DIMACS Challenge and the QPLIB, and solve some of them (for the first time) to optimality.}, language = {en} } @article{LeKoch2023, author = {Le, Thi Thai and Koch, Thorsten}, title = {Effect of inertia force on the interface stability of a tangential-velocity discontinuity in porous media}, volume = {169}, journal = {International Journal of Multiphase Flow}, doi = {10.1016/j.ijmultiphaseflow.2023.104612}, year = {2023}, abstract = {The present study investigates the stability of a tangential-velocity discontinuity in porous media during the withdrawing and injecting processes of natural gases from and into an underground gas storage. The focus is placed on analyzing the impact of inertia forces on the interface stability using the Forchheimer equations. Other publications have relied primarily on Darcy's law to describe flow stability in porous media. However, Darcy's law only adequately describes flows in which viscous forces dominate over inertia forces. As the flow rate increases, the significance of inertia forces becomes more pronounced, and Darcy's law becomes insufficient for considering such flows. Our findings indicate that even a slight consideration of the inertia effect leads to permanent destabilization of the discontinuity interface, regardless of the fluid viscosity or the Mach number. In contrast, when the inertia effect is neglected, the interface is stabilized across the entire Mach number range if the fluid viscosity is strong enough.}, language = {en} } @article{ChenKochZakiyevaetal.2023, author = {Chen, Ying and Koch, Thorsten and Zakiyeva, Nazgul and Liu, Kailiang and Xu, Zhitong and Chen, Chun-houh and Nakano, Junji and Honda, Keisuke}, title = {Article's scientific prestige: Measuring the impact of individual articles in the web of science}, volume = {17}, journal = {Journal of Informetrics}, number = {1}, doi = {10.1016/j.joi.2023.101379}, pages = {101379}, year = {2023}, abstract = {We performed a citation analysis on the Web of Science publications consisting of more than 63 million articles and 1.45 billion citations on 254 subjects from 1981 to 2020. We proposed the Article's Scientific Prestige (ASP) metric and compared this metric to number of citations (\#Cit) and journal grade in measuring the scientific impact of individual articles in the large-scale hierarchical and multi-disciplined citation network. In contrast to \#Cit, ASP, that is computed based on the eigenvector centrality, considers both direct and indirect citations, and provides steady-state evaluation cross different disciplines. We found that ASP and \#Cit are not aligned for most articles, with a growing mismatch amongst the less cited articles. While both metrics are reliable for evaluating the prestige of articles such as Nobel Prize winning articles, ASP tends to provide more persuasive rankings than \#Cit when the articles are not highly cited. The journal grade, that is eventually determined by a few highly cited articles, is unable to properly reflect the scientific impact of individual articles. The number of references and coauthors are less relevant to scientific impact, but subjects do make a difference.}, language = {en} } @inproceedings{TurnerBertholdBesanconetal.2023, author = {Turner, Mark and Berthold, Timo and Besan{\c{c}}on, Mathieu and Koch, Thorsten}, title = {Cutting Plane Selection with Analytic Centers and Multiregression}, volume = {13884}, booktitle = {Integration of Constraint Programming, Artificial Intelligence, and Operations Research. CPAIOR 2023.}, publisher = {Springer}, doi = {10.1007/978-3-031-33271-5_4}, pages = {52 -- 68}, year = {2023}, abstract = {Cutting planes are a crucial component of state-of-the-art mixed-integer programming solvers, with the choice of which subset of cuts to add being vital for solver performance. We propose new distance-based measures to qualify the value of a cut by quantifying the extent to which it separates relevant parts of the relaxed feasible set. For this purpose, we use the analytic centers of the relaxation polytope or of its optimal face, as well as alternative optimal solutions of the linear programming relaxation. We assess the impact of the choice of distance measure on root node performance and throughout the whole branch-and-bound tree, comparing our measures against those prevalent in the literature. Finally, by a multi-output regression, we predict the relative performance of each measure, using static features readily available before the separation process. Our results indicate that analytic center-based methods help to significantly reduce the number of branch-and-bound nodes needed to explore the search space and that our multiregression approach can further improve on any individual method.}, language = {en} } @misc{TurnerBertholdBesanconetal.2022, author = {Turner, Mark and Berthold, Timo and Besan{\c{c}}on, Mathieu and Koch, Thorsten}, title = {Cutting Plane Selection with Analytic Centers and Multiregression}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-89065}, year = {2022}, abstract = {Cutting planes are a crucial component of state-of-the-art mixed-integer programming solvers, with the choice of which subset of cuts to add being vital for solver performance. We propose new distance-based measures to qualify the value of a cut by quantifying the extent to which it separates relevant parts of the relaxed feasible set. For this purpose, we use the analytic centers of the relaxation polytope or of its optimal face, as well as alternative optimal solutions of the linear programming relaxation. We assess the impact of the choice of distance measure on root node performance and throughout the whole branch-and-bound tree, comparing our measures against those prevalent in the literature. Finally, by a multi-output regression, we predict the relative performance of each measure, using static features readily available before the separation process. Our results indicate that analytic center-based methods help to significantly reduce the number of branch-and-bound nodes needed to explore the search space and that our multiregression approach can further improve on any individual method.}, language = {en} } @article{KochBertholdPedersenetal.2022, author = {Koch, Thorsten and Berthold, Timo and Pedersen, Jaap and Vanaret, Charlie}, title = {Progress in mathematical programming solvers from 2001 to 2020}, volume = {10}, journal = {EURO Journal on Computational Optimization}, doi = {10.1016/j.ejco.2022.100031}, pages = {100031}, year = {2022}, abstract = {This study investigates the progress made in lp and milp solver performance during the last two decades by comparing the solver software from the beginning of the millennium with the codes available today. On average, we found out that for solving lp/milp, computer hardware got about 20 times faster, and the algorithms improved by a factor of about nine for lp and around 50 for milp, which gives a total speed-up of about 180 and 1,000 times, respectively. However, these numbers have a very high variance and they considerably underestimate the progress made on the algorithmic side: many problem instances can nowadays be solved within seconds, which the old codes are not able to solve within any reasonable time.}, language = {en} } @misc{BleyGleixnerKochetal.2009, author = {Bley, Andreas and Gleixner, Ambros and Koch, Thorsten and Vigerske, Stefan}, title = {Comparing MIQCP solvers to a specialised algorithm for mine production scheduling}, organization = {ZIB}, issn = {1438-0064}, doi = {10.1007/978-3-642-25707-0}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11507}, number = {09-32}, year = {2009}, abstract = {In this paper we investigate the performance of several out-of-the box solvers for mixed-integer quadratically constrained programmes (MIQCPs) on an open pit mine production scheduling problem with mixing constraints. We compare the solvers BARON, Couenne, SBB, and SCIP to a problem-specific algorithm on two different MIQCP formulations. The computational results presented show that general-purpose solvers with no particular knowledge of problem structure are able to nearly match the performance of a hand-crafted algorithm.}, language = {en} } @misc{RehfeldtFujisawaKochetal.2022, author = {Rehfeldt, Daniel and Fujisawa, Katsuki and Koch, Thorsten and Nakao, Masahiro and Shinano, Yuji}, title = {Computing single-source shortest paths on graphs with over 8 trillion edges}, issn = {1438-0064}, doi = {10.12752/8818}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-88180}, year = {2022}, abstract = {This paper introduces an implementation for solving the single-source shortest path problem on distributed-memory machines. It is tailored to power-law graphs and scales to trillions of edges. The new implementation reached 2nd and 10th place in the latest Graph500 benchmark in June 2022 and handled the largest and second-largest graphs among all participants.}, language = {en} } @misc{VuLitzelKoch2025, author = {Vu, Thi Huong and Litzel, Ida and Koch, Thorsten}, title = {Similarity-based fuzzy clustering scientific articles: potentials and challenges from mathematical and computational perspectives}, arxiv = {http://arxiv.org/abs/2506.04045}, year = {2025}, abstract = {Fuzzy clustering, which allows an article to belong to multiple clusters with soft membership degrees, plays a vital role in analyzing publication data. This problem can be formulated as a constrained optimization model, where the goal is to minimize the discrepancy between the similarity observed from data and the similarity derived from a predicted distribution. While this approach benefits from leveraging state-of-the-art optimization algorithms, tailoring them to work with real, massive databases like OpenAlex or Web of Science -- containing about 70 million articles and a billion citations -- poses significant challenges. We analyze potentials and challenges of the approach from both mathematical and computational perspectives. Among other things, second-order optimality conditions are established, providing new theoretical insights, and practical solution methods are proposed by exploiting the problem's structure. Specifically, we accelerate the gradient projection method using GPU-based parallel computing to efficiently handle large-scale data.}, language = {en} } @misc{VuKoch2025, author = {Vu, Thi Huong and Koch, Thorsten}, title = {Clustering scientific publications: lessons learned through experiments with a real citation network}, arxiv = {http://arxiv.org/abs/2505.18180}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-100418}, year = {2025}, abstract = {Clustering scientific publications helps uncover research structures within bibliographic databases. Graph-based methods such as spectral, Louvain, and Leiden clustering are commonly used due to their ability to model citation networks. However, their effectiveness can diminish when applied to real-world data. This study evaluates these clustering algorithms on a citation graph of about 700,000 articles and 4.6 million citations from the Web of Science. The results show that while scalable methods like Louvain and Leiden perform efficiently, their default settings often yield poor partitioning. Meaningful outcomes require careful parameter tuning, especially for large networks with uneven structures, including a dense core and loosely connected papers. These findings highlight practical lessons about the challenges of large-scale data, method selection and tuning based on specific structures of bibliometric clustering tasks.}, language = {en} } @article{KempkeRehfeldtKoch2025, author = {Kempke, Nils-Christian and Rehfeldt, Daniel and Koch, Thorsten}, title = {A Massively Parallel Interior-Point-Method for Arrowhead Linear Programs}, journal = {SIAM Journal on Scientific Computing}, arxiv = {http://arxiv.org/abs/2412.07731}, year = {2025}, language = {en} } @article{RongChenMaetal.2025, author = {Rong, Guoyang and Chen, Ying and Ma, Feicheng and Koch, Thorsten}, title = {Exploring Interdisciplinary Research Trends through Critical Years for Interdisciplinary Citation}, volume = {19}, journal = {Journal of Informetrics}, number = {4}, doi = {10.1016/j.joi.2025.101726}, year = {2025}, language = {en} } @misc{PedersenLindnerRehfeldtetal.2025, author = {Pedersen, Jaap and Lindner, Niels and Rehfeldt, Daniel and Koch, Thorsten}, title = {Integrated Wind Farm Design: Optimizing Turbine Placement and Cable Routing with Wake Effects}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-99218}, year = {2025}, abstract = {An accelerated deployment of renewable energy sources is crucial for a successful transformation of the current energy system, with wind energy playing a key role in this transition. This study addresses the integrated wind farm layout and cable routing problem, a challenging nonlinear optimization problem. We model this problem as an extended version of the Quota Steiner Tree Problem (QSTP), optimizing turbine placement and network connectivity simultaneously to meet specified expansion targets. Our proposed approach accounts for the wake effect - a region of reduced wind speed induced by each installed turbine - and enforces minimum spacing between turbines. We introduce an exact solution framework in terms of the novel Quota Steiner Tree Problem with interference (QSTPI). By leveraging an interference-based splitting strategy, we develop an advanced solver capable of tackling large-scale problem instances. The presented approach outperforms generic state-of-the-art mixed integer programming solvers on our dataset by up to two orders of magnitude. Moreover, we demonstrate that our integrated method significantly reduces the costs in contrast to a sequential approach. Thus, we provide a planning tool that enhances existing planning methodologies for supporting a faster and cost-efficient expansion of wind energy.}, language = {en} } @misc{KempkeKuntKatamishetal.2025, author = {Kempke, Nils-Christian and Kunt, Tim and Katamish, Bassel and Vanaret, Charlie and Sasanpour, Shima and Clarner, Jan-Patrick and Koch, Thorsten}, title = {Developing heuristic solution techniques for large-scale unit commitment models}, issn = {1438-0064}, arxiv = {http://arxiv.org/abs/2502.19012}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-99555}, year = {2025}, abstract = {Shifting towards renewable energy sources and reducing carbon emissions necessitate sophisticated energy system planning, optimization, and extension. Energy systems optimization models (ESOMs) often form the basis for political and operational decision-making. ESOMs are frequently formulated as linear (LPs) and mixed-integer linear (MIP) problems. MIPs allow continuous and discrete decision variables. Consequently, they are substantially more expressive than LPs but also more challenging to solve. The ever-growing size and complexity of ESOMs take a toll on the computational time of state-of-the-art commercial solvers. Indeed, for large-scale ESOMs, solving the LP relaxation -- the basis of modern MIP solution algorithms -- can be very costly. These time requirements can render ESOM MIPs impractical for real-world applications. This article considers a set of large-scale decarbonization-focused unit commitment models with expansion decisions based on the REMix framework (up to 83 million variables and 900,000 discrete decision variables). For these particular instances, the solution to the LP relaxation and the MIP optimum lie close. Based on this observation, we investigate the application of relaxation-enforced neighborhood search (RENS), machine learning guided rounding, and a fix-and-propagate (FP) heuristic as a standalone solution method. Our approach generated feasible solutions 20 to 100 times faster than GUROBI, achieving comparable solution quality with primal-dual gaps as low as 1\% and up to 35\%. This enabled us to solve numerous scenarios without lowering the quality of our models. For some instances that Gurobi could not solve within two days, our FP method provided feasible solutions in under one hour.}, language = {en} } @inproceedings{StomporZittelKochetal.2025, author = {Stompor, Tomasz and Zittel, Janina and Koch, Thorsten and Rusch, Beate}, title = {Fully Algorithmic Librarian: Large-Scale Citation Experiments}, booktitle = {20th International Society of Scientometrics and Informetrics Conference 2025, June 23-27, 2025 | Yerevan, Armenia}, doi = {10.51408/issi2025_204}, year = {2025}, abstract = {The Fully Algorithmic Librarian (FAN) project explores application scenarios for algorithmic-intelligence(AI)-supported methods in academic libraries as central institutions for research support. To this end, the study builds on two algorithmic approaches for analyzing large-scale citation networks. A comparison of Web of Science (WoS) and OpenAlex structures using the PageRank algorithm reveals key differences. Additionally, a multi-label clustering technique designed for large-scale citation networks accounts for disciplinary variations in publication practices.}, language = {en} } @article{ChenKochPengetal.2025, author = {Chen, Ying and Koch, Thorsten and Peng, Hanqui and Zhang, Hongrui}, title = {Benchmarking of Quantum and Classical Computing in Large-Scale Dynamic Portfolio Optimization Under Market Frictions}, arxiv = {http://arxiv.org/abs/2502.05226}, year = {2025}, language = {en} } @article{ChenXuKoch2020, author = {Chen, Ying and Xu, Xiuqin and Koch, Thorsten}, title = {Day-ahead high-resolution forecasting of natural gas demand and supply in Germany with a hybrid model}, volume = {262}, journal = {Applied Energy}, number = {114486}, doi = {https://doi.org/10.1016/j.apenergy.2019.114486}, year = {2020}, abstract = {As the natural gas market is moving towards short-term planning, accurate and robust short-term forecasts of the demand and supply of natural gas is of fundamental importance for a stable energy supply, a natural gas control schedule, and transport operation on a daily basis. We propose a hybrid forecast model, Functional AutoRegressive and Convolutional Neural Network model, based on state-of-the-art statistical modeling and artificial neural networks. We conduct short-term forecasting of the hourly natural gas flows of 92 distribution nodes in the German high-pressure gas pipeline network, showing that the proposed model provides nice and stable accuracy for different types of nodes. It outperforms all the alternative models, with an improved relative accuracy up to twofold for plant nodes and up to fourfold for municipal nodes. For the border nodes with rather flat gas flows, it has an accuracy that is comparable to the best performing alternative model.}, language = {en} } @article{TjusilaBesanconTurneretal.2024, author = {Tjusila, Gennesaret and Besan{\c{c}}on, Mathieu and Turner, Mark and Koch, Thorsten}, title = {How Many Clues To Give? A Bilevel Formulation For The Minimum Sudoku Clue Problem}, volume = {54}, journal = {Operations Research Letters}, doi = {10.1016/j.orl.2024.107105}, pages = {107105}, year = {2024}, abstract = {It has been shown that any 9 by 9 Sudoku puzzle must contain at least 17 clues to have a unique solution. This paper investigates the more specific question: given a particular completed Sudoku grid, what is the minimum number of clues in any puzzle whose unique solution is the given grid? We call this problem the Minimum Sudoku Clue Problem (MSCP). We formulate MSCP as a binary bilevel linear program, present a class of globally valid inequalities, and provide a computational study on 50 MSCP instances of 9 by 9 Sudoku grids. Using a general bilevel solver, we solve 95\% of instances to optimality, and show that the solution process benefits from the addition of a moderate amount of inequalities. Finally, we extend the proposed model to other combinatorial problems in which uniqueness of the solution is of interest.}, language = {en} } @misc{TurnerBertholdBesanconetal.2023, author = {Turner, Mark and Berthold, Timo and Besan{\c{c}}on, Mathieu and Koch, Thorsten}, title = {Branching via Cutting Plane Selection: Improving Hybrid Branching}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-91120}, year = {2023}, abstract = {Cutting planes and branching are two of the most important algorithms for solving mixed-integer linear programs. For both algorithms, disjunctions play an important role, being used both as branching candidates and as the foundation for some cutting planes. We relate branching decisions and cutting planes to each other through the underlying disjunctions that they are based on, with a focus on Gomory mixed-integer cuts and their corresponding split disjunctions. We show that selecting branching decisions based on quality measures of Gomory mixed-integer cuts leads to relatively small branch-and-bound trees, and that the result improves when using cuts that more accurately represent the branching decisions. Finally, we show how the history of previously computed Gomory mixed-integer cuts can be used to improve the performance of the state-of-the-art hybrid branching rule of SCIP. Our results show a 4\% decrease in solve time, and an 8\% decrease in number of nodes over affected instances of MIPLIB 2017.}, language = {en} } @article{LeFukumotoKoch2024, author = {Le, Thi Thai and Fukumoto, Yasuhide and Koch, Thorsten}, title = {Linear stability of a simple shear layer between two parallel streams in a shallow water flow}, volume = {493}, journal = {Physics Letters A}, publisher = {Elservier}, doi = {10.1016/j.physleta.2023.129264}, year = {2024}, abstract = {The stability of shear layers in fluid flows is a crucial factor in forming vortices and jets and plays a fundamental role in the development of turbulence. Such shear layer instabilities are ubiquitous in natural phenomena, such as atmospheric and oceanic flows, contributing to the formation of weather systems and predicting tsunamis. This study specifically focuses on the stability of a shear layer sandwiched between two semi-infinite layers within a two-dimensional flow. The velocity profile of the shear layer is assumed to be linearly dependent on the vertical coordinate, while the velocity of the other layers remains uniform with differing strengths. The effect of viscosity and surface tension is ignored to simplify the analysis. The shallow water equations are used to analyze the interface stability of the shear layer, and the resulting dispersion relation between wave frequency and other wave characteristics is obtained. This relation incorporates Whittaker functions and their first derivatives and is used to derive appropriate limits corresponding to various physical conditions. Our study thus contributes to a deeper understanding of the stability of shear layers and their implications for natural phenomena.}, language = {en} } @misc{TjusilaBesanconTurneretal.2023, author = {Tjusila, Gennesaret and Besancon, Mathieu and Turner, Mark and Koch, Thorsten}, title = {How Many Clues To Give? A Bilevel Formulation For The Minimum Sudoku Clue Problem}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-90902}, year = {2023}, abstract = {It has been shown that any 9 by 9 Sudoku puzzle must contain at least 17 clues to have a unique solution. This paper investigates the more specific question: given a particular completed Sudoku grid, what is the minimum number of clues in any puzzle whose unique solution is the given grid? We call this problem the Minimum Sudoku Clue Problem (MSCP). We formulate MSCP as a binary bilevel linear program, present a class of globally valid inequalities, and provide a computational study on 50 MSCP instances of 9 by 9 Sudoku grids. Using a general bilevel solver, we solve 95\\% of instances to optimality, and show that the solution process benefits from the addition of a moderate amount of inequalities. Finally, we extend the proposed model to other combinatorial problems in which uniqueness of the solution is of interest.}, language = {en} } @misc{PrauseHoppmannBaumDefournyetal.2020, author = {Prause, Felix and Hoppmann-Baum, Kai and Defourny, Boris and Koch, Thorsten}, title = {The Maximum Diversity Assortment Selection Problem}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-81039}, year = {2020}, abstract = {In this paper, we introduce the Maximum Diversity Assortment Selection Problem (MADASS), which is a generalization of the 2-dimensional Cutting Stock Problem (2CSP). Given a set of rectangles and a rectangular container, the goal of 2CSP is to determine a subset of rectangles that can be placed in the container without overlapping, i.e., a feasible assortment, such that a maximum area is covered. In MADASS, we need to determine a set of feasible assortments, each of them covering a certain minimum threshold of the container, such that the diversity among them is maximized. Thereby, diversity is defined as minimum or average normalized Hamming-Distance of all assortment pairs. The MADASS Problem was used in the 11th AIMMS-MOPTA Competition in 2019. The methods we describe in this article and the computational results won the contest. In the following, we give a definition of the problem, introduce a mathematical model and solution approaches, determine upper bounds on the diversity, and conclude with computational experiments conducted on test instances derived from the 2CSP literature.}, language = {en} }