TY - GEN A1 - Bosse, Hartwig A1 - Grötschel, Martin A1 - Henk, Martin T1 - Polynomial Inequalities Representing Polyhedra N2 - Our main result is that every $n$-dimensional polytope can be described by at most $2n-1$ polynomial inequalities and, moreover, these polynomials can explicitly be constructed. For an $n$-dimensional pointed polyhedral cone we prove the bound $2n-2$ and for arbitrary polyhedra we get a constructible representation by $2n$ polynomial inequalities. T3 - ZIB-Report - 04-53 KW - polyhedra and polytopes KW - semi-algebraic sets KW - polyhedral combinatorics KW - polynomial inequalities KW - stability index Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8284 ER - TY - GEN A1 - Grötschel, Martin A1 - Martin, Alexander A1 - Weismantel, Robert T1 - Optimum Path Packing on Wheels: The Consecutive Case N2 - We show that, given a wheel with nonnegative edge lengths and pairs of terminals located on the wheel's outer cycle such that the terminal pairs are in consecutive order, then a path packing, i.~e., a collection of edge disjoint paths connecting the given terminal pairs, of minimum length can be found in strongly polynomial time. Moreover, we exhibit for this case a system of linear inequalities that provides a complete and nonredundant description of the path packing polytope, which is the convex hull of all incidence vectors of path packings and their supersets. T3 - ZIB-Report - SC-95-31 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1976 ER - TY - GEN A1 - Grötschel, Martin A1 - Martin, Alexander A1 - Weismantel, Robert T1 - The Steiner Tree Packing Problem in VLSI-Design. N2 - In this paper we describe several versions of the routing problem arising in VLSI design and indicate how the Steiner tree packing problem can be used to model these problems mathematically. We focus on switchbox routing problems and provide integer programming formulations for routing in the knock-knee and in the Manhattan model. We give a brief sketch of cutting plane algorithms that we developed and implemented for these two models. We report on computational experiments using standard test instances. Our codes are able to determine optimum solutions in most cases, and in particular, we can show that some of the instances have no feasible solution if Manhattan routing is used instead of knock-knee routing. T3 - ZIB-Report - SC-94-02 Y1 - 1994 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1328 ER - TY - GEN A1 - Grötschel, Martin A1 - Martin, Alexander A1 - Weismantel, Robert T1 - Optimum Path Packing on Wheels: The Noncrossing Case. N2 - We show that, given a wheel with nonnegative edge lengths and pairs of terminals located on the wheel's outer cycle such that no two terminal pairs cross, then a path packing, i.~e.,a collection of edge disjoint paths connecting the given terminal pairs, of minimum length can be found in strongly polynomial time. Moreover, we exhibit for this case a system of linear inequalities that provides a complete and nonredundant description of the path packing polytope, which is the convex hull of all incidence vectors of path packings and their supersets. T3 - ZIB-Report - SC-93-26 Y1 - 1993 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1225 ER - TY - GEN A1 - Grötschel, Martin A1 - Martin, Alexander A1 - Weismantel, Robert T1 - Packing Steiner Trees: Polyhedral Investigations. N2 - Let $G=(V,E)$ be a graph and $T\subseteq V$ be a node set. We call an edge set $S$ a Steiner tree with respect to $T$ if $S$ connects all pairs of nodes in $T$. In this paper we address the following problem, which we call the weighted Steiner tree packing problem. Given a graph $G=(V,E)$ with edge weights $w_e$, edge capacities $c_e, e \in E,$ and node sets $T_1,\ldots,T_N$, find edge sets $S_1,\ldots,S_N$ such that each $S_k$ is a Steiner tree with respect to $T_k$, at most $c_e$ of these edge sets use edge $e$ for each $e\in E$, and such that the sum of the weights of the edge sets is minimal. Our motivation for studying this problem arises from the routing problem in VLSI-design, where given sets of points have to be connected by wires. We consider the Steiner tree packing Problem from a polyhedral point of view and define an appropriate polyhedron, called the Steiner tree packing polyhedron. The goal of this paper is to (partially) describe this polyhedron by means of inequalities. It turns out that, under mild assumptions, each inequality that defines a facet for the (single) Steiner tree polyhedron can be lifted to a facet-defining inequality for the Steiner tree packing polyhedron. The main emphasis of this paper lies on the presentation of so-called joint inequalities that are valid and facet-defining for this polyhedron. Inequalities of this kind involve at least two Steiner trees. The classes of inequalities we have found form the basis of a branch & cut algorithm. This algorithm is described in our companion paper SC 92-09. T3 - ZIB-Report - SC-92-08 KW - cutting planes KW - Facets KW - Packing KW - polyhedron KW - Steiner tree Y1 - 1992 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-786 ER - TY - GEN A1 - Grötschel, Martin A1 - Martin, Alexander A1 - Weismantel, Robert T1 - Packing Steiners Trees: A Cutting Plane Algorithm and Computational Results. N2 - In this paper we describe a cutting plane algorithm for the Steiner tree packing problem. We use our algorithm to solve some switchbox routing problems of VLSI-design and report on our computational experience. This includes a brief discussion of separation algorithms, a new LP-based primal heuristic and implementation details. The paper is based on the polyhedral theory for the Steiner tree packing polyhedron developed in our companion paper SC 92-8 and meant to turn this theory into an algorithmic tool for the solution of practical problems. T3 - ZIB-Report - SC-92-09 Y1 - 1992 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-792 ER - TY - GEN A1 - Ferreira, Carlos E. A1 - Grötschel, Martin A1 - Kiefl, Stefan A1 - Krispenz, Ludwig A1 - Martin, Alexander A1 - Weismantel, Robert T1 - Some Integer Programs Arising in the Design of Main Frame Computers. N2 - In this paper we describe and discuss a problem that arises in the (global) design of a main frame computer. The task is to assign certain functional units to a given number of so called multi chip modules or printed circuit boards taking into account many technical constraints and minimizing a complex objective function. We describe the real world problem. A thorough mathematical modelling of all aspects of this problem results in a rather complicated integer program that seems to be hopelessly difficult -- at least for the present state of integer programming technology. We introduce several relaxations of the general model, which are also $NP$-hard, but seem to be more easily accessible. The mathematical relations between the relaxations and the exact formulation of the problem are discussed as well. T3 - ZIB-Report - SC-92-25 Y1 - 1993 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-957 ER - TY - GEN A1 - Grötschel, Martin A1 - Martin, Alexander A1 - Weismantel, Robert T1 - Routing in Grid Graphs by Cutting Planes. N2 - {\def\N{{\cal N}} \def\R{\hbox{\rm I\kern-2pt R}} \def\MN{{\rm I\kern-2pt N}} In this paper we study the following problem, which we call the weighted routing problem. Let be given a graph $G=(V,E)$ with non-negative edge weights $w_e\in\R_+$ and integer edge capacities $c_e\in\MN$ and let $\N=\{T_1,\ldots,T_N\}$, $N\ge 1$, be a list of node sets. The weighted routing problem consists in finding edge sets $S_1,\ldots,S_N$ such that, for each $k\in\{1,\ldots,N\}$, the subgraph $(V(S_k),S_k)$ contains an $[s,t]$-path for all $s,t\in T_k$, at most $c_e$ of these edge sets use edge $e$ for each $e\in E$, and such that the sum of the weights of the edge sets is minimal. Our motivation for studying this problem arises from the routing problem in VLSI-design, where given sets of points have to be connected by wires. We consider the weighted routing problem from a polyhedral point of view. We define an appropriate polyhedron and try to (partially) describe this polyhedron by means of inequalities. We briefly sketch our separation algorithms for some of the presented classes of inequalities. Based on these separation routines we have implemented a branch and cut algorithm. Our algorithm is applicable to an important subclass of routing problems arising in VLSI-design, namely to problems where the underlying graph is a grid graph and the list of node sets is located on the outer face of the grid. We report on our computational experience with this class of problem instances.} T3 - ZIB-Report - SC-92-26 KW - routing in VLSI-design KW - Steiner tree KW - Steiner tree packing KW - cutting plane algorithm Y1 - 1992 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-967 ER - TY - GEN A1 - Grötschel, Martin A1 - Martin, Alexander A1 - Weismantel, Robert T1 - Packing Steiner Trees: Further Facets. N2 - In this paper we continue the investigations in [GMW92a] for the \def\sbppo{Steiner tree packing polyhedron} \sbppo. We present several new classes of valid inequalities and give sufficient (and necessary) conditions for these inequalities to be facet-defining. It is intended to incorporate these inequalities into an existing cutting plane algorithm that is applicable to practical problems arising in the design of electronic circuits. T3 - ZIB-Report - SC-93-01 Y1 - 1992 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-975 ER - TY - GEN A1 - Grötschel, Martin A1 - Martin, Alexander A1 - Weismantel, Robert T1 - Packing Steiner Trees: Separation Algorithms. N2 - In this paper we investigate separation problems for classes of inequalities valid for the polytope associated with the Steiner tree packing problem, a problem that arises, e.~g., in VLSI routing. The separation problem for Steiner partition inequalities is ${\cal N}\hskip-2pt{\cal P}$-hard in general. We show that it can be solved in polynomial time for those instances that come up in switchbox routing. Our algorithm uses dynamic programming techniques. These techniques are also applied to the much more complicated separation problem for alternating cycle inequalities. In this case we can compute in polynomial time, given some point $y$, a lower bound for the gap $\alpha-a^Ty$ over all alternating cycle inequalities $a^Tx\ge\alpha$. This gives rise to a very effective separation heuristic. A by-product of our algorithm is the solution of a combinatorial optimization problem that is interesting in its own right: Find a shortest path in a graph where the ``length'' of a path is its usual length minus the length of its longest edge. T3 - ZIB-Report - SC-93-02 Y1 - 1992 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-984 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Eisenblätter, Andreas A1 - Grötschel, Martin A1 - Martin, Alexander T1 - Frequency Assignment in Cellular Phone Networks N2 - We present a graph-theoretic model for the \emph{frequency assignment problem} in Cellular Phone Networks: Obeying several technical and legal restrictions, frequencies have to be assigned to transceivers so that interference is as small as possible. This optimization problem is NP-hard. Good approximation cannot be guaranteed, unless P = NP. We describe several assignment heuristics. These heuristics are simple and not too hard to implement. We give an assessment of the heuristics' efficiency and practical usefulness. For this purpose, typical instances of frequency assignment problems with up to 4240 transceivers and 75 frequencies of a German cellular phone network operator are used. The results are satisfying from a practitioner's point of view. The best performing heuristics were integrated into a network planning system used in practice. T3 - ZIB-Report - SC-97-35 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3044 ER - TY - GEN A1 - Grötschel, Martin A1 - Henk, Martin T1 - On the Representation of Polyhedra by Polynomial Inequalities N2 - A beautiful result of Bröcker and Scheiderer on the stability index of basic closed semi-algebraic sets implies, as a very special case, that every $d$-dimensional polyhedron admits a representation as the set of solutions of at most $d(d+1)/2$ polynomial inequalities. Even in this polyhedral case, however, no constructive proof is known, even if the quadratic upper bound is replaced by any bound depending only on the dimension. Here we give, for simple polytopes, an explicit construction of polynomials describing such a polytope. The number of used polynomials is exponential in the dimension, but in the 2- and 3-dimensional case we get the expected number $d(d+1)/2$. T3 - ZIB-Report - 02-15 KW - polyhedra and polytopes KW - semialgebraic sets KW - polyhedral combinatorics KW - polynomial inequalities Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6826 ER - TY - GEN A1 - Bosse, Hartwig A1 - Grötschel, Martin A1 - Henk, Martin T1 - Polynomial Inequalities Representing Polyhedra N2 - Our main result is that every n-dimensional polytope can be described by at most (2n-1) polynomial inequalities and, moreover, these polynomials can explicitly be constructed. For an n-dimensional pointed polyhedral cone we prove the bound 2n-2 and for arbitrary polyhedra we get a constructible representation by 2n polynomial inequalities. T3 - ZIB-Report - 03-25 KW - polyhedra and polytopes KW - semi-algebraic sets KW - polyhedral combinatorics KW - polynomial inequalities KW - stability index Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7473 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Eisenblätter, Andreas A1 - Grötschel, Martin A1 - Martin, Alexander T1 - The Orientation Model for Frequency Assignment Problems N2 - Mobile telecommunication systems establish a large number of communication links with a limited number of available frequencies; reuse of the same or adjacent frequencies on neighboring links causes interference. The task to find an assignment of frequencies to channels with minimal interference is the frequency assignment problem. The frequency assignment problem is usually treated as a graph coloring problem where the number of colors is minimized, but this approach does not model interference minimization correctly. We give in this paper a new integer programming formulation of the frequency assignment problem, the orientation model, and develop a heuristic two-stage method to solve it. The algorithm iteratively solves an outer and an inner optimization problem. The outer problem decides for each pair of communication links which link gets the higher frequency and leads to an acyclic subdigraph problem with additional longest path restrictions. The inner problem to find an optimal assignment respecting an orientation leads to a min-cost flow problem. T3 - ZIB-Report - TR-98-01 KW - Minimum-Cost Flow Problems KW - Cellular Radio Telephone Systems KW - Frequency Assignment Problem KW - Integer Programming Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5627 ER - TY - JOUR A1 - Grötschel, Martin A1 - Martin, Alexander A1 - Weismantel, Robert T1 - Packing Steiner trees: Polyhedral Investigations JF - Mathematical Programming, Series A Y1 - 1996 U6 - https://doi.org/10.1007/BF02592085 VL - 72 IS - 2 SP - 101 EP - 123 ER - TY - JOUR A1 - Grötschel, Martin A1 - Martin, Alexander A1 - Weismantel, Robert T1 - Optimum Path Packing on Wheels JF - Computers and Mathematics with Applications Y1 - 1996 VL - 31 IS - 11 SP - 23 EP - 35 PB - Elsevier ER - TY - JOUR A1 - Bosse, Hartwig A1 - Grötschel, Martin A1 - Henk, Martin T1 - Polynomial inequalities representing polyhedra JF - Mathematical Programming Y1 - 2005 U6 - https://doi.org/10.1007/s10107-004-0563-2 VL - 103 IS - 1 SP - 35 EP - 44 ER - TY - JOUR A1 - Grötschel, Martin A1 - Henk, Martin T1 - The Representation of Polyhedra by Polynomial Inequalities JF - Discrete & Computational Geometry Y1 - 2003 U6 - https://doi.org/10.1007/s00454-003-0782-y VL - 29 IS - 4 SP - 485 EP - 504 ER - TY - JOUR A1 - Borndörfer, Ralf A1 - Eisenblätter, Andreas A1 - Grötschel, Martin A1 - Martin, Alexander T1 - Frequency assignment in cellular phone networks JF - Annals of Operations Research Y1 - 1998 U6 - https://doi.org/10.1023/A:1018908907763 VL - 76 SP - 73 EP - 93 PB - J. C. Baltzer AG, Science Publishers ER - TY - JOUR A1 - Grötschel, Martin A1 - Martin, Alexander A1 - Weismantel, Robert T1 - Packing Steiner trees: a cutting plane algorithm and computational results JF - Mathematical Programming, Series A Y1 - 1996 U6 - https://doi.org/10.1007/BF02592086 VL - 72 IS - 2 SP - 125 EP - 145 ER - TY - JOUR A1 - Grötschel, Martin A1 - Martin, Alexander A1 - Weismantel, Robert T1 - The Steiner tree packing problem in VLSI design JF - Mathematical Programming Y1 - 1997 VL - 78 IS - 2 SP - 265 EP - 281 ER - TY - CHAP A1 - Grötschel, Martin A1 - Martin, Alexander A1 - Weismantel, Robert ED - Rinaldi, Giovanni ED - Wolsey, Laurence T1 - Routing in grid graphs by cutting planes T2 - Integer Programming and Combinatorial Optimization. Proceedings of a Conference held at Centro Ettore Majorana, Erice, Italy, April 29 - May 1, 1993 Y1 - 1993 SP - 447 EP - 461 PB - Librarian CORE CY - Louvain-la-Neuve ER - TY - JOUR A1 - Ferreira, Carlos A1 - Grötschel, Martin A1 - Martin, Alexander A1 - Weismantel, Robert A1 - Kiefl, Stefan A1 - Krispenz, Ludwig T1 - Some Integer Programs Arising in the Design of Main Frame Computers JF - ZOR - Methods and Models of Operations Research Y1 - 1993 VL - 38 IS - 1 SP - 77 EP - 100 ER - TY - JOUR A1 - Grötschel, Martin A1 - Martin, Alexander A1 - Weismantel, Robert T1 - Routing in Grid Graphs by Cutting Planes JF - ZOR - Mathematical Methods of Operations Research Y1 - 1995 VL - 41 IS - 3 SP - 255 EP - 275 ER - TY - JOUR A1 - Grötschel, Martin A1 - Martin, Alexander A1 - Weismantel, Robert T1 - Packing Steiner Trees: Separation Algorithms JF - SIAM Journal on Discrete Mathematics Y1 - 1996 U6 - https://doi.org/10.1137/S0895480193258716 VL - 9 IS - 2 SP - 233 EP - 257 ER - TY - JOUR A1 - Grötschel, Martin A1 - Martin, Alexander A1 - Weismantel, Robert T1 - Packing Steiner Trees: Further Facets JF - European Journal of Combinatorics Y1 - 1996 U6 - https://doi.org/10.1006/eujc.1996.0004 VL - 17 IS - 1 SP - 39 EP - 52 ER - TY - CHAP A1 - Borndörfer, Ralf A1 - Eisenblätter, Andreas A1 - Grötschel, Martin A1 - Martin, Alexander T1 - Stable-Set and Other Techniques for Frequency Assignment Problems T2 - Anais da I Oficina Nacional em Problemas de Corte & Empacotamento Y1 - 1996 SP - 17 EP - 21 PB - Universidade de São Paulo, Instituto de Matemática e Estatística ER - TY - GEN A1 - Grötschel, Martin A1 - Lügger, Joachim T1 - Veränderungen der Publikations- und Kommunikationswelt: Open Access, Google, etc. T1 - Changes in the world of publishing and communication: Open Access, Google etc. N2 - Vorbemerkung Beim Schreiben dieses Artikels, der auf Veränderungen der Kommunikations- und Publikationstechniken und ihre Bedeutung hinweist, ist uns mehr als je zuvor bewusst geworden, wie beschränkt das Medium Papier ist. Es gibt z. B. keine Hyperlinks, durch die man unmittelbar das Erwähnte erleben oder überprüfen kann. Ein schneller Wechsel vom Wort zum Bild, zum Ton oder Video ist nicht möglich. Wer will schon lange URLs abtippen und Medienbrüche erleiden? Wir haben uns daher entschlossen, eine textidentische Version dieses Artikels mit allen URLs – sie liegt Ihnen hier vor – elektronisch anzubieten und in der für die "Gegenworte" (BBAW) gekürzten Fassung nur durch [URL] anzudeuten, dass der Leser an dieser Stelle einfach in der elektronischen Version einen Klick ins Internet machen sollte. Und damit sind wir bereits mitten im Thema. T3 - ZIB-Report - 09-10 KW - Publikations- und Kommunikationstechniken KW - Open Access KW - techniques of publishing and communication KW - Open Access Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11223 SN - 1438-0064 ER - TY - GEN A1 - Grötschel, Martin A1 - Lügger, Joachim T1 - E-Learning and the Working Mathematician N2 - Eigentlich war der erste Autor nur zu einem Grußwort zur Tagung „GML² 2009 - Grundfragen Multi¬medialen Lehrens und Lernens“ eingeladen. Daraus wurde ein E-Learning-bezogener Vortrag, der – basierend auf Erfahrungen im Fach Mathematik – einen kritischen Blick auf die E-Learning-Szene in Deutschland wirft und diese mit entsprechenden Aktivitäten weltweit vergleicht. Dies ist die in seinen mathematischen Teilen gekürzte, in den E-Learning-Anteilen ein wenig erweiterte schriftliche Fassung des Vortrags. Der Artikel stammt nicht von E-Learning-Spezialisten sondern von Personen, die sich seit fast zwanzig Jahren mit elektronischer Information und Kommunikation (kurz: IuK) – insbesondere in der Mathematik – beschäftigen. Nach einer Definition von Michael Kerres kennzeichnet der Begriff E-Learning (electronic learning – elektronisch unterstütztes Lernen) alle Formen von Lernen, bei denen digitale Medien für die Präsentation und Distribution von Lernmaterialien und/oder zur Unterstützung zwischenmenschlicher Kommunikation zum Einsatz kommen, siehe z.B. http://de.wikipedia.org/wiki/E-Learning. IuK und E-Learning haben nach dieser Begriffsbildung viele Berührungspunkte. Deswegen wagen wir es, unsere positiven und negativen Erfahrungen im Bereich IuK in diesem Eröffnungsvortrag zu berichten, einige Entwicklungslinien zu vergleichen und eine eigene Kurzversion der Definition von E-Learning (besser E-Teaching and -Learning) voranzustellen: „Lehren und Lernen mit Unterstützung elektronischer Hilfsmittel“. T3 - ZIB-Report - 09-29 KW - E-Learning KW - elektronisches Lehren und Lernen KW - e-learning KW - electronic learning Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11455 UR - http://www.gml-2009.de/media/tagungsbandgml2009.pdf SN - 1438-0064 ER - TY - GEN A1 - Grötschel, Martin T1 - Das Problem mit der Komplexität: P = NP? N2 - Was Komplexität ist, weiß niemand so richtig. In vielen Wissenschaftsgebieten wird der Begriff Komplexität verwendet, überall mit etwas anderer Bedeutung. Mathematik und Informatik hab en eine eigene Theorie hierzu entwickelt: die Komplexitätstheorie. Sie stellt zwar grundlegende Begriffe bereit, aber leider sind die meisten wichtigen Fragestellungen noch ungelöst. Diese kurze Einführung konzentriert sich auf einen speziellen, aber bedeutenden Aspekt der Theorie: Lösbarkeit von Problemen in deterministischer und nichtdeterministischer polynomialer Zeit. Hinter der für Uneingeweihte etwas kryptischen Frage "P = NP?" verbirgt sich das derzeit wichtigste Problem der Komplexitätstheorie. Anhand dieser Fragestellung werden einige Aspekte der Theorie erläutert und formell erklärt, was "P = NP?" bedeutet. Es geht nicht nur um komplizierte algorithmische Mathematik und Informatik, sondern um grundsätzliche Fragen unserer Lebensumwelt. Kann man vielleicht beweisen, dass es für viele Probleme unseres Alltags keine effizienten Lösungsmethoden gibt? T3 - ZIB-Report - 05-58 KW - Komplexität KW - Komplexitätstheorie Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8916 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Grötschel, Martin A1 - Pfetsch, Marc T1 - A Column-Generation Approach to Line Planning in Public Transport N2 - The line planning problem is one of the fundamental problems in strategic planning of public and rail transport. It consists in finding lines and corresponding frequencies in a transport network such that a given travel demand can be satisfied. There are (at least) two objectives. The transport company wishes to minimize operating costs, the passengers want to minimize travel times. We propose a n ew multi-commodity flow model for line planning. Its main features, in comparison to existing models, are that the passenger paths can be freely routed and that the lines are generated dynamically. We discuss properties of this model and investigate its complexity. Results with data for the city of Potsdam, Germany, are reported. T3 - ZIB-Report - 05-18 KW - line planning KW - column generation Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8522 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Grötschel, Martin A1 - Pfetsch, Marc T1 - Public Transport to the fORe! N2 - Can OR methods help the public transport industry to break even? The article gives evidence that there exist significant potentials in this direction, which can be harnessed by a combination of modern mathematical methods and local planning knowledge. Many of the planning steps in public transport are classical combinatorial problems, which can be solved in unprecedented size and quality due the rapid progress in large-scale optimization. Three examples on vehicle scheduling, duty scheduling, and integrated vehicle and duty scheduling illustrate the level that has been reached and the improvements that can be achieved today. Extensions of such methods to further questions of strategic, online, and market-oriented planning are currently investigated. In this way, OR can make a significant contribution to answer the basic but extremely difficult question ``What is a good public transport network?. T3 - ZIB-Report - 05-22 KW - Public transport KW - vehicle scheduling KW - duty scheduling Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8562 UR - http://www.lionhrtpub.com/orms/orms-4-06/frtransport.html ER - TY - GEN A1 - Grötschel, Martin A1 - Roggenbuck, Robert A1 - Sperber, Wolfram T1 - Entwicklung und Aufbau des Internet-Dienstes "Math&Industry" N2 - Die Anwendung mathematischer Methoden und Verfahren wird immer mehr zur Voraussetzung innovativer Produkte und Dienstleistungen. Um neue Produkte und Dienstleistungen zu entwickeln, müssen die Produktions- und technologischen Prozesse mathematisch modelliert, beschrieben und optimiert werden. Diesen Umstand Rechnung tragend, hat das Bundesministerium für Bildung und Forschung (BMBF) 1993 begonnen, den Einsatz mathematischer Verfahren und Methoden in der Mathematik über ein spezielles Mathematikprogramm zu fördern. Inzwischen hat die vierte Förderperiode des Mathematikprogramms begonnen. \par Das Medium Internet und insbesondere das WWW sind für die Sichtbarkeit und Transparenz wissenschaftlicher Resultate in den letzten zehn Jahren immer wichtiger geworden. Wer nicht im Web "'sichtbar"' ist, läuft Gefahr, nicht wahrgenommen zu werden. Intention und Ziel des durchgeführten Projekts war es, ein Konzept für eine qualitativ hochwertige und umfassende Darstellung des BMBF Mathematikprogramms, insbesondere der in den Projekten erzielten Ergebnisse, zu entwickeln und zu realisieren und damit den Stellenwert und die Akzeptanz mathematischer Forschung in der Gesellschaft zu festigen und den Wissenstransfer zwischen mathematischer Forschung sowie Forschung und Entwicklung in der Wirtschaft und dem Dienstleistungsbereich zu fördern. T3 - ZIB-Report - 05-23 KW - Internet services KW - distributed systems KW - XML KW - RDF Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8573 ER - TY - GEN A1 - Grötschel, Martin T1 - Schnelle Rundreisen: Das Travelling Salesman-Problem N2 - Das Travelling-Salesman-Problem (TSP) ist das am intensivsten untersuchte kombinatorische Optimierungsproblem. In diesem Abschnitt wird eine Einführung in das TSP gegeben. Es werden Problemstellungen erläutert, Anwendungen skizziert und einige Schwierigkeiten bei der korrekten Modellierung der Zielfunktion dargelegt. Es ist gar nicht so klar, was in einem konkreten Problem die wirkliche Entfernung ist. Exakte und approximative Lösungsverfahren werden an Beispielen skizziert, und es wird angedeutet, dass man, obwohl TSPs zu den theoretisch schweren Problemen zählen, in der Praxis TSPs von atemberaubender Größe lösen kann. T3 - ZIB-Report - 05-57 KW - Travelling-Salesman-Problem KW - TSP KW - Rundreisen KW - optimale Tour Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8901 ER - TY - GEN A1 - Grötschel, Martin A1 - Hinrichs, Hinrich A1 - Schröer, Klaus A1 - Tuchscherer, Andreas T1 - Ein gemischt-ganzzahliges lineares Optimierungsmodell für ein Laserschweißproblem im Karosseriebau N2 - Wir beschäftigen uns mit dem Problem der Betriebsplanung von Laserschweißrobotern im Karosseriebau. Gegeben ist eine Menge von Schweißnähten, die innerhalb einer Fertigungszelle an einem Karosserieteil gefertigt werden müssen. Die Schweißnähte werden durch mehrere parallel betriebene Roboter bearbeitet. Die Aufgabe besteht darin, für jeden Roboter eine Reihenfolge und eine zeitliche Koordinierung seiner Bewegungen zu finden, so dass alle Schweißnähte innerhalb der Taktzeit der Fertigungszelle bearbeitet werden und so wenig Laserquellen wie möglich eingesetzt werden. Dabei müssen einige Nebenbedingungen berücksichtigt werden. Für dieses spezielle Schweißproblem haben wir eine Formulierung als gemischt-ganzzahliges lineares Programm entwickelt, welches sich für die untersuchten praktischen Fälle sehr schnell lösen lässt. T3 - ZIB-Report - 06-21 KW - Laserschweißen KW - Gemischt-ganzzahlige Optimierung Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9141 ER - TY - GEN A1 - Achterberg, Tobias A1 - Grötschel, Martin A1 - Koch, Thorsten T1 - Software for Teaching Modeling of Integer Programming Problems N2 - Modern applications of mathematical programming must take into account a multitude of technical details, business demands, and legal requirements. Teaching the mathematical modeling of such issues and their interrelations requires real-world examples that are well beyond the toy sizes that can be tackled with the student editions of most commercial software packages. We present a new tool, which is freely available for academic use including complete source code. It consists of an algebraic modeling language and a linear mixed integer programming solver. The performance and features of the tool are in the range of current state-of-the-art commercial tools, though not in all aspects as good as the best ones. Our tool does allow the execution and analysis of large real-world instances in the classroom and can therefore enhance the teaching of problem solving issues. Teaching experience has been gathered and practical usability was tested in classes at several universities and a two week intensive block course at TU Berlin. The feedback from students and teachers has been very positive. T3 - ZIB-Report - 06-23 KW - Integer Programming KW - Modelling KW - MIP-Solver KW - Algebraic Modelling Languages KW - Teaching Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9176 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Grötschel, Martin A1 - Löbel, Andreas ED - Aigner, Martin ED - Behrends, Ehrhard T1 - Der schnellste Weg zum Ziel T2 - Alles Mathematik Y1 - 2000 U6 - https://doi.org/10.1007/978-3-322-96366-6_5 SP - 45 EP - 76 PB - Vieweg Verlag CY - Braunschweig/Wiesbaden ER - TY - CHAP A1 - Borndörfer, Ralf A1 - Grötschel, Martin A1 - Jaeger, Ulrich ED - Grötschel, Martin ED - Lucas, Klaus ED - Mehrmann, Volker T1 - Planungsprobleme im öffentlichen Verkehr T2 - PRODUKTIONSFAKTOR MATHEMATIK – Wie Mathematik Technik und Wirtschaft bewegt Y1 - 2008 UR - http://opus.kobv.de/zib/volltexte/2008/1103/ SP - 127 EP - 153 PB - acatech – Deutsche Akademie der Technikwissenschaften und Springer ER - TY - CHAP A1 - Borndörfer, Ralf A1 - Grötschel, Martin A1 - Pfetsch, Marc ED - Hickman, Mark ED - Mirchandani, Pitu ED - Voß, Stefan T1 - Models for Line Planning in Public Transport T2 - Computer-aided Systems in Public Transport (CASPT 2004) Y1 - 2008 U6 - https://doi.org/10.1007/978-3-540-73312-6_18 VL - 600 SP - 363 EP - 378 PB - Springer-Verlag ER - TY - THES A1 - Grötschel, Martin T1 - Der Satz von Frobenius für konvergente Potenzreihen Y1 - 1973 ER - TY - JOUR A1 - Grötschel, Martin A1 - Padberg, Manfred T1 - Partial linear characterizations of the asymmetric travelling salesman polytope JF - Mathematical Programming Y1 - 1975 VL - 8 SP - 378 EP - 381 ER - TY - CHAP A1 - Grötschel, Martin ED - Kohlas, Jürg ED - Seifert, Olaf ED - Zimmermann, H. T1 - Über Graphen mit Kreisen, die gegebene Wege enthalten T2 - Proceedings in Operations Research Y1 - 1976 VL - 5 SP - 119 EP - 121 PB - Physica-Verlag CY - Würzburg ER - TY - CHAP A1 - Grötschel, Martin ED - Hammer, Peter ED - Johnson, Ellis ED - Korte, Bernhard ED - Nemhauser, George T1 - Graphs with cycles containing given paths T2 - Studies in Integer Programming Y1 - 1977 VL - 1 SP - 233 EP - 245 PB - North-Holland ER - TY - BOOK A1 - Grötschel, Martin T1 - Polyedrische Charakterisierungen kombinatorischer Optimierungsprobleme T3 - Mathematical Systems in Economics Y1 - 1977 SN - 3-445-01537-6 VL - 36 PB - Verlag Anton Hain CY - Meisenheim am Glan ER - TY - CHAP A1 - Grötschel, Martin T1 - The monotone 2-matching polytope on a complete graph T2 - Operations Research Verfahren XXIV, Symposium in Mannheim Y1 - 1977 SP - 72 EP - 84 ER - TY - CHAP A1 - Grötschel, Martin T1 - Facetten von Matroid-Polytopen T2 - Operations Research Verfahren XXV, Symposium Heidelberg 1976 Y1 - 1977 SP - 306 EP - 313 ER - TY - CHAP A1 - Grötschel, Martin A1 - Korte, Bernhard ED - Albach, Horst ED - Helmstädter, Ernst ED - Henn, Rudolf T1 - Bemerkungen zur Optimierung ökonomischer Modelle T2 - Quantitative Wirtschaftsforschung, Willhelm Krelle zum 60. Geburtstag Y1 - 1977 SP - 457 EP - 467 PB - J. C. B. Mohr CY - Tübingen ER - TY - JOUR A1 - Grötschel, Martin A1 - Padberg, Manfred T1 - Lineare Charakterisierungen von Travelling Salesman Problemen JF - Zeitschrift für Operations Research, Series A Y1 - 1977 VL - 21 IS - 1 SP - 33 EP - 64 PB - Physica-Verlag ER - TY - JOUR A1 - Grötschel, Martin T1 - Hypo-Hamiltonian Facets of the Symmetric Travelling Salesman Polytope JF - Zeitschrift für Angewandte Mathematik und Mechanik Y1 - 1978 VL - 58 SP - T469 EP - T471 ER - TY - CHAP A1 - Grötschel, Martin A1 - Padberg, Manfred ED - Henn, Rudolf ED - Korte, Bernhard ED - Oettli, W. T1 - On the symmetric travelling salesman problem T2 - Optimization and operations research, Proc. Workshop, Bonn 1977 Y1 - 1978 VL - 157 SP - 105 EP - 115 PB - Springer ER - TY - JOUR A1 - Grötschel, Martin A1 - von Randow, Rabe T1 - A property of continuous unbounded algorithms JF - Mathematical Programming Y1 - 1978 VL - 14 SP - 373 EP - 377 ER - TY - JOUR A1 - Grötschel, Martin T1 - On minimal strong blocks JF - Journal of Graph Theory Y1 - 1979 VL - 3 SP - 213 EP - 219 ER - TY - CHAP A1 - Grötschel, Martin ED - Collatz, Lothar ED - Meinardus, G. ED - Wetterling (Hrsg.), W. T1 - Strong blocks and the optimum branching problem T2 - Numerische Methoden bei graphentheoretischen und kombinatorischen Problemen, Bd. 2, Tagung in Oberwolfach 1978 Y1 - 1979 VL - 46 SP - 112 EP - 121 PB - Birkhäuser CY - Basel ER - TY - JOUR A1 - Grötschel, Martin A1 - Harary, Frank T1 - The Graphs for which All Strong Orientations Are Hamiltonian JF - Journal of Graph Theory Y1 - 1979 VL - 3 SP - 221 EP - 223 ER - TY - JOUR A1 - Grötschel, Martin A1 - Padberg, Manfred T1 - On the symmetric travelling salesman problem I JF - Mathematical Programming Y1 - 1979 VL - 16 SP - 265 EP - 280 ER - TY - JOUR A1 - Grötschel, Martin A1 - Padberg, Manfred T1 - On the symmetric travelling salesman problem II JF - Mathematical Programming Y1 - 1979 VL - 16 SP - 281 EP - 302 ER - TY - JOUR A1 - Grötschel, Martin T1 - On the symmetric travelling salesman problem JF - Mathematical Programming Study Y1 - 1980 VL - 12 SP - 61 EP - 77 PB - North-Holland ER - TY - JOUR A1 - Grötschel, Martin T1 - On the monotone symmetric travelling salesman problem JF - Mathematics of Operations Research Y1 - 1980 VL - 5 IS - 2 SP - 285 EP - 292 ER - TY - CHAP A1 - Grötschel, Martin A1 - Wakabayashi, Yoshiko ED - Bodendiek, Rainer ED - Henn, Rudolf T1 - Composition of facets of the clique partitioning polytope T2 - Topics in combinatorics and graph theory. Essays in honour of Gerhard Ringel Y1 - 1990 SP - 271 EP - 284 PB - Physica-Verlag CY - Heidelberg ER - TY - CHAP A1 - Grötschel, Martin A1 - Jünger, Michael A1 - Reinelt, Gerhard ED - van Hemmen, J. ED - Morgenstern, I. T1 - Calculating Exact Ground States of Spin Glasses T2 - Proceedings of the Heidelberg Colloquium on “Glassy Dynamics”, Lecture Notes in Physics 275 Y1 - 1987 SP - 325 EP - 353 PB - Springer ER - TY - JOUR A1 - Barahona, Francisco A1 - Grötschel, Martin A1 - Jünger, Michael A1 - Reinelt, Gerhard T1 - An application of combinatorial optimization to statistical physics and circuit layout design JF - Operations Research Y1 - 1988 VL - 36 IS - 3 SP - 493 EP - 513 ER - TY - BOOK A1 - Grötschel, Martin A1 - Lovász, Lászlo A1 - Schrijver, Alexander T1 - Geometric Algorithms and Combinatorial Optimization T3 - Algorithms and Combinatorics Y1 - 1988 SN - 3-540-13624-X, 0-387-13624-X (U.S.) VL - 2 PB - Springer ER - TY - JOUR A1 - Grötschel, Martin A1 - Jünger, Michael A1 - Reinelt, Gerhard T1 - Via Minimization with Pin Preassignments and Layer Preference JF - ZAMM – Zeitschrift für Angewandte Mathematik und Mechanik Y1 - 1989 VL - 69 IS - 11 SP - 393 EP - 399 ER - TY - JOUR A1 - Grötschel, Martin A1 - Truemper, Klaus T1 - Decomposition and Optimization over Cycles in Binary Matroids JF - Journal of Combinatorial Theory, Series B Y1 - 1989 VL - 46 IS - 3 SP - 306 EP - 337 ER - TY - JOUR A1 - Grötschel, Martin A1 - Truemper, Klaus T1 - Master Polytopes for Cycles of Binary Matroids JF - Linear Algebra and its Applications Y1 - 1989 VL - 114/115 SP - 523 EP - 540 ER - TY - JOUR A1 - Grötschel, Martin A1 - Wakabayashi, Yoshiko T1 - A cutting plane algorithm for a clustering problem JF - Mathematical Programming, Series B Y1 - 1989 VL - 45 SP - 59 EP - 96 PB - Mathematical Programming, Series B ER - TY - CHAP A1 - Ascheuer, Norbert A1 - Escudero, Laureano A1 - Grötschel, Martin A1 - Stoer, Mechthild ED - Kannan, Ravi ED - Pulleyblank, William T1 - On Identifying in Polynomial Time Violated Subtour Elimination and Precedence Forcing Constraints for the Sequential Ordering Problem T2 - Integer Programming and Combinatorial Optimization, Proceeedings of a Conference held at the University of Waterloo, May 28-30, 1990 Y1 - 1990 SP - 19 EP - 28 PB - University of Waterloo Press ER - TY - JOUR A1 - Grötschel, Martin A1 - Monma, Clyde T1 - Integer Polyhedra Arising from Certain Network Design Problems with Connectivity Constraints JF - SIAM Journal on Discrete Mathematics Y1 - 1990 VL - 3 IS - 4 SP - 502 EP - 523 ER - TY - JOUR A1 - Grötschel, Martin A1 - Wakabayashi, Yoshiko T1 - Facets of the clique partitioning polytope JF - Mathematical Programming, Series A Y1 - 1990 VL - 47 IS - 3 SP - 367 EP - 387 ER - TY - JOUR A1 - Grötschel, Martin A1 - Jünger, Michael A1 - Reinelt, Gerhard T1 - Optimal triangulation of large real world input-output matrices JF - Statistische Hefte Y1 - 1984 VL - 25 SP - 261 EP - 295 ER - TY - CHAP A1 - Grötschel, Martin A1 - Lovász, Lászlo A1 - Schrijver, Alexander ED - Pulleyblank, William T1 - Geometric Methods in Combinatorial Optimization T2 - Progress in Combinatorial Optimization, (Conference Waterloo/Ontario 1982) Y1 - 1984 SP - 167 EP - 183 PB - Academic Press ER - TY - CHAP A1 - Grötschel, Martin A1 - Lovász, Lászlo A1 - Schrijver, Alexander ED - Berge, Claude ED - Chvátal, Vašek T1 - Polynomial algorithms for perfect graphs T2 - Topics on perfect graphs Y1 - 1984 VL - 21 SP - 325 EP - 356 PB - North-Holland ER - TY - JOUR A1 - Grötschel, Martin A1 - Lovász, Lászlo A1 - Schrijver, Alexander T1 - Corrigendum to our paper “The ellipsoid method and its consequences in combinatorial optimization” JF - Combinatorica Y1 - 1984 VL - 4 IS - 4 SP - 291 EP - 295 ER - TY - JOUR A1 - Grötschel, Martin A1 - Nemhauser, George T1 - A polynomial algorithm for the max-cut problem on graphs without long odd cycles JF - Mathematical Programming Y1 - 1984 VL - 29 IS - 1 SP - 28 EP - 40 PB - North-Holland ER - TY - CHAP A1 - Grötschel, Martin A1 - Wakabayashi, Yoshiko ED - Cottle, Richard ED - Kelmanson, Milton ED - Korte, Bernhard T1 - Constructions of hypotraceable digraphs T2 - Mathematical Programming. Proceedings of the International Congress on Mathematical Programming, Rio de Janeiro, Brazil, 6-8 April 1981 Y1 - 1984 SP - 147 EP - 175 PB - North-Holland ER - TY - JOUR A1 - Barahona, Francisco A1 - Grötschel, Martin A1 - Mahjoub, Ali T1 - Facets of the Bipartite Subgraph Polytope JF - Mathematics of Operations Research Y1 - 1985 VL - 10 SP - 340 EP - 358 ER - TY - JOUR A1 - Grötschel, Martin A1 - Holland, Olaf T1 - Solving matching problems with linear programming JF - Mathematical Programming Y1 - 1985 VL - 33 IS - 3 SP - 243 EP - 259 ER - TY - CHAP A1 - Grötschel, Martin A1 - Jünger, Michael A1 - Reinelt, Gerhard ED - Rival, Ivan T1 - Acyclic subdigraphs and linear orderings T2 - Graphs and Order. The Role of Graphs in the Theory of Ordered Sets and its Applications, Proceedings of NATO ASI, Ser. C 147, Banff/Canada 1984 Y1 - 1985 SP - 217 EP - 264 PB - D. Reidel Publishing Company ER - TY - JOUR A1 - Grötschel, Martin A1 - Jünger, Michael A1 - Reinelt, Gerhard T1 - On the acyclic subgraph polytope JF - Mathematical Programming Y1 - 1985 VL - 33 IS - 1 SP - 28 EP - 42 PB - North-Holland ER - TY - JOUR A1 - Grötschel, Martin A1 - Jünger, Michael A1 - Reinelt, Gerhard T1 - Facets of the linear ordering polytope JF - Mathematical Programming Y1 - 1985 VL - 33 IS - 1 SP - 43 EP - 60 PB - North-Holland ER - TY - CHAP A1 - Grötschel, Martin A1 - Padberg, Manfred ED - Lawler, Eugene ED - Lenstra, Jan ED - Kan, Alexander ED - Shmoys, David T1 - Polyhedral theory T2 - The Traveling Salesman Problem. A Guided Tour of Combinatorial Optimization Y1 - 1985 SP - 251 EP - 306 PB - Wiley ER - TY - CHAP A1 - Padberg, Manfred A1 - Grötschel, Martin ED - Lawler, Eugene ED - Lenstra, Jan ED - Kan, Alexander ED - Shmoys, David T1 - Polyhedral computations T2 - The Traveling Salesman Problem. A Guided Tour of Combinatorial Optimization Y1 - 1985 SP - 307 EP - 360 PB - Wiley ER - TY - RPRT A1 - Barahona, Francisco A1 - Grötschel, Martin T1 - The Travelling Salesman Problem for Graphs not Contractible to K_5 - e Y1 - 1985 IS - 77 PB - Mathematisches Institut, Universität Augsburg ER - TY - JOUR A1 - Barahona, Francisco A1 - Grötschel, Martin T1 - On the Cycle Polytope of a Binary Matroid JF - Journal of Combinatorial Theory, Series B Y1 - 1986 VL - 40 SP - 40 EP - 62 ER - TY - CHAP A1 - Grötschel, Martin ED - Kerber, Adalbert T1 - Polyedrische Kombinatorik und Schnittebenenverfahren, Tagungsbericht 2. Sommerschule Diskrete Strukturen, Bayreuth 1985 T2 - Diskrete Strukturen, algebraische Methoden und Anwendungen Y1 - 1986 VL - 21 SP - 69 EP - 100 PB - Institut für Mathematik, Universität Bayreuth ER - TY - CHAP A1 - Grötschel, Martin ED - Krelle, Wilhelm T1 - Fortschritte in der polyedrischen Kombinatorik T2 - Ökonomische Prognose-, Entscheidungs- und Gleichgewichtsmodelle, Abschlußberichte des SFB 21 Y1 - 1986 SP - 341 EP - 367 PB - Verlag Chemie CY - Weinheim ER - TY - JOUR A1 - Grötschel, Martin A1 - Lovász, Lászlo A1 - Schrijver, Alexander T1 - Relaxations of Vertex Packing JF - Journal of Combinatorial Theory, Series B Y1 - 1986 VL - 40 SP - 330 EP - 343 ER - TY - JOUR A1 - Grötschel, Martin A1 - Pulleyblank, William T1 - Clique tree inequalities and the symmetric travelling salesman problem JF - Mathematics of Operations Research Y1 - 1986 VL - 11 IS - 4 SP - 537 EP - 569 ER - TY - JOUR A1 - Grötschel, Martin A1 - Holland, Olaf T1 - A Cutting Plane Algorithm for Minimum Perfect 2-Matchings JF - Computing Y1 - 1987 VL - 39 SP - 327 EP - 344 ER - TY - JOUR A1 - Grötschel, Martin A1 - Thomassen, Carsten A1 - Wakabayashi, Yoshiko T1 - Hypotraceable digraphs JF - Journal of Graph Theory Y1 - 1980 VL - 4 SP - 377 EP - 381 ER - TY - JOUR A1 - Grötschel, Martin A1 - Wakabayashi, Yoshiko T1 - Hypohamiltonian digraphs JF - Methods of Operations Research Y1 - 1980 VL - 36 SP - 99 EP - 119 ER - TY - BOOK A1 - Bachem, Achim A1 - Butz, Lothar A1 - Grötschel, Martin A1 - (Bearb.), Rainer A1 - Leitung), Bernhard T1 - Input-Output-Analyse bei unternehmensgrößenspezifischen Fragestellungen Y1 - 1981 PB - Institut für Mittelstandsforschung CY - Bonn ER - TY - JOUR A1 - Bachem, Achim A1 - Grötschel, Martin T1 - Homogenization of polyhedra JF - Methods of Operations Research Y1 - 1981 VL - 40 SP - 215 EP - 219 ER - TY - CHAP A1 - Bachem, Achim A1 - Grötschel, Martin ED - König, Heinz ED - Korte, Bernhard ED - Ritter, Klaus T1 - Characterizations of adjacency of faces of polyhedra T2 - Mathematical Programming at Oberwolfach Y1 - 1981 VL - 14 SP - 1 EP - 22 PB - North-Holland ER - TY - JOUR A1 - Grötschel, Martin A1 - Lovász, Lászlo A1 - Schrijver, Alexander T1 - The ellipsoid method and its consequences in combinatorial optimization JF - Combinatorica Y1 - 1981 VL - 1 IS - 2 SP - 169 EP - 197 ER - TY - JOUR A1 - Grötschel, Martin A1 - Lovász, Lászlo A1 - Schrijver, Alexander T1 - The ellipsoid method and its consequences in combinatorial optimization JF - IIASA Collab. Proceedings Ser. CP-81-S1 Y1 - 1981 SP - 511 EP - 546 ER - TY - JOUR A1 - Grötschel, Martin A1 - Pulleyblank, William T1 - Weakly bipartite graphs and the max-cut problem JF - Operations Research Letters Y1 - 1981 VL - 1 IS - 1 SP - 23 EP - 27 ER - TY - JOUR A1 - Grötschel, Martin A1 - Wakabayashi, Yoshiko T1 - On the structure of the monotone asymmetric travelling salesman polytope I JF - Discrete Mathematics Y1 - 1981 VL - 34 SP - 43 EP - 59 ER - TY - JOUR A1 - Grötschel, Martin A1 - Wakabayashi, Yoshiko T1 - On the structure of the monotone asymmetric travelling salesman polytope II JF - Mathematical Programming Study Y1 - 1981 VL - 14 SP - 77 EP - 97 ER - TY - CHAP A1 - Bachem, Achim A1 - Grötschel, Martin ED - Korte, Bernhard T1 - New aspects of polyhedral theory T2 - Modern Applied Mathematics Y1 - 1982 SP - 51 EP - 106 PB - North-Holland ER - TY - CHAP A1 - Grötschel, Martin ED - Korte, Bernhard T1 - Approaches to Hard Combinatorial Optimization Problems T2 - Modern Applied Mathematics, Optimization and Operations Research, Lect. Summer Sch., Bonn 1979 Y1 - 1982 SP - 437 EP - 515 PB - North-Holland ER -