@misc{KosterZymolka2000, author = {Koster, Arie M.C.A. and Zymolka, Adrian}, title = {Stable Multi-Sets}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6047}, number = {00-36}, year = {2000}, abstract = {In this paper we introduce a generalization of stable sets: stable multi-sets. A stable multi-set is an assignment of integers to the vertices of a graph, such that specified bounds on vertices and edges are not exceeded. In case all vertex and edge bounds equal one, stable multi-sets are equivalent to stable sets. For the stable multi-set problem, we derive reduction rules and study the associated polytope. We state necessary and sufficient conditions for the extreme points of the linear relaxation to be integer. These conditions generalize the conditions for the stable set polytope. Moreover, the classes of odd cycle and clique inequalities for stable sets are generalized to stable multi-sets and conditions for them to be facet defining are determined. The study of stable multi-sets is initiated by optimization problems in the field of telecommunication networks. Stable multi-sets emerge as an important substructure in the design of optical networks.}, language = {en} } @misc{KochMartinVoss2000, author = {Koch, Thorsten and Martin, Alexander and Voß, Stefan}, title = {SteinLib: An Updated Library on Steiner Tree Problems in Graphs}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6056}, number = {00-37}, year = {2000}, abstract = {In this paper we present the {\em SteinLib}, a library of data sets for the Steiner tree problem in graphs. This library extends former libraries on Steiner tree problems by many new interesting and difficult instances, most of them arising from real-world applications. We give a survey on the difficulty of these problem instances by giving references to state-of-the-art software packages that were the first or are currently among the best to solve these instances.}, language = {en} } @misc{KosterZymolka2003, author = {Koster, Arie M.C.A. and Zymolka, Adrian}, title = {Polyhedral Investigations on Stable Multi-Sets}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7324}, number = {03-10}, year = {2003}, abstract = {Stable multi-sets are an evident generalization of the well-known stable sets. As integer programs, they constitute a general structure which allows for a wide applicability of the results. Moreover, the study of stable multi-sets provides new insights to well-known properties of stable sets. In this paper, we continue our investigations started in [{\sl Koster and Zymolka 2002}] and present results of three types: on the relation to other combinatorial problems, on the polyhedral structure of the stable multi-set polytope, and on the computational impact of the polyhedral results. First of all, we embed stable multi-sets in a framework of generalized set packing problems and point out several relations. The second part discusses properties of the stable multi-set polytope. We show that the vertices of the linear relaxation are half integer and have a special structure. Moreover, we strengthen the conditions for cycle inequalities to be facet defining, show that the separation problem for these inequalities is polynomial time solvable, and discuss the impact of chords in cycles. The last result allows to interpret cliques as cycles with many chords. The paper is completed with a computational study to the practical importance of the cycle inequalities. The computations show that the performance of state-of-the-art integer programming solvers can be improved significantly by including these inequalities.}, language = {en} } @misc{EisenblaetterKosterWallbaumetal.2002, author = {Eisenbl{\"a}tter, Andreas and Koster, Arie M.C.A. and Wallbaum, Randolf and Wess{\"a}ly, Roland}, title = {Load Balancing in Signaling Transfer Points}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7179}, number = {02-50}, year = {2002}, abstract = {Signaling is crucial to the operation of modern telecommunication networks. A breakdown in the signaling infrastructure typically causes customer service failures, incurs revenue losses, and hampers the company image. Therefore, the signaling network has to be highest reliability and survivability. This in particular holds for the routers in such a network, called \textit{signaling transfer points\/} (STPs). The robustness of an STP can be improved by equally distributing the load over the internal processing units. Several constraints have to be taken into account. The load of the links connected to a processing unit changes over time introducing an imbalance of the load. In this paper, we show how integer linear programming can be applied to reduce the imbalance within an STP, while keeping the number of changes small. Two alternative models are presented. Computational experiments validate the integer programming approach in practice. The GSM network operator E-Plus saves substantial amounts of time and money by employing the proposed approach.}, language = {en} } @misc{ZymolkaKosterWessaely2002, author = {Zymolka, Adrian and Koster, Arie M.C.A. and Wess{\"a}ly, Roland}, title = {Transparent optical network design with sparse wavelength conversion}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7017}, number = {02-34}, year = {2002}, abstract = {We consider the design of transparent optical networks from a practical perspective. Network operators aim at satisfying the communication demands at minimum cost. Such an optimization involves three interdependent planning issues: the dimensioning of the physical topology, the routing of lightpaths, and the wavelength assignment. Further topics include the reliability of the configuration and sparse wavelength conversion for efficient use of the capacities. In this paper, we investigate this extensive optical network design task. Using a flexible device-based model, we present an integer programming formulation that supports greenfield planning as well as expansion planning on top of an existing network. As solution method, we propose a suitable decomposition approach that separates the wavelength assignment from the dimensioning and routing. Our method in particular provides a lower bound on the total cost which allows to rate the solution quality. Computational experiments on realistic networks approve the solution approach to be appropriate.}, language = {en} } @misc{BorndoerferEisenblaetterGroetscheletal.1998, author = {Bornd{\"o}rfer, Ralf and Eisenbl{\"a}tter, Andreas and Gr{\"o}tschel, Martin and Martin, Alexander}, title = {The Orientation Model for Frequency Assignment Problems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5627}, number = {TR-98-01}, year = {1998}, abstract = {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.}, language = {en} } @misc{AscheuerFischettiGroetschel1999, author = {Ascheuer, Norbert and Fischetti, Matteo and Gr{\"o}tschel, Martin}, title = {Solving the Asymmetric Travelling Salesman Problem with Time Windows by Branch-and-Cut}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4191}, number = {SC-99-31}, year = {1999}, abstract = {Many optimization problems have several equivalent mathematical models. It is often not apparent which of these models is most suitable for practical computation, in particular, when a certain application with a specific range of instance sizes is in focus. Our paper addresses the Asymmetric Travelling Salesman Problem with time windows (ATSP-TW) from such a point of view. The real--world application we aim at is the control of a stacker crane in a warehouse. We have implemented codes based on three alternative integer programming formulations of the ATSP-TW and more than ten heuristics. Computational results for real-world instances with up to 233 nodes are reported, showing that a new model presented in a companion paper outperforms the other two models we considered --- at least for our special application --- and that the heuristics provide acceptable solutions.}, language = {en} } @misc{Groetschel2000, author = {Gr{\"o}tschel, Martin}, title = {Verkehrsplanung: Bessere L{\"o}sungen mit Mathematik}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4439}, number = {SC-99-54}, year = {2000}, abstract = {Anwendungen der Mathematik in der Verkehrs- und Transporttechnologie haben eine große und bedeutende Tradition. Nat{\"u}rlich wurden die ersten Fahrzeuge mit der ingenieurm{\"a}ßigen Methode von Versuch, Irrtum und Verbesserung entworfen. Aber schon sehr bald kamen mathematische Berechnungen hinzu, mit denen mechanische Eigenschaften von Fahrzeugteilen ermittelt und zum Teil optimiert wurden. Die hierzu erforderliche Mathematik wurde in diesem Jahrhundert zu einem m{\"a}chtigen Werkzeugkasten ausgebaut. Mit diesem kann man heute z.B. hocheffiziente Motoren mit geringem Schadstoffausstoß entwerfen, aerodynamisch g{\"u}nstige Fahrzeugprofile ermitteln und Flugzeugfl{\"u}gel berechnen, die die gew{\"u}nschte Last sicher und mit geringem Treibstoffaufwand tragen. Die Mathematik unterst{\"u}tzt die Technologie des Verkehrs beginnend bei globalen Designfragen bis hin zur Spezifizierung von Materialeigenschaften kleinster Bauteile; sie berechnet mit hoher Pr{\"a}zision energieoptimale Bahnen von Raumflugk{\"o}rpern oder zeitoptimale Trajektorien f{\"u}r Flugzeuge, steuert automatische Roboteranlagen oder innerbetriebliche Transportsysteme.}, language = {de} } @misc{MarchandMartinWeismanteletal.1999, author = {Marchand, Hugues and Martin, Alexander and Weismantel, Robert and Wolsey, Laurence}, title = {Cutting Planes in Integer and Mixed Integer Programming}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4339}, number = {SC-99-44}, year = {1999}, abstract = {This survey presents cutting planes that are useful or potentially useful in solving mixed integer programs. Valid inequalities for i) general integer programs, ii) problems with local structure such as knapsack constraints, and iii) problems with 0-1 coefficient matrices, such as set packing, are examined in turn. Finally the use of valid inequalities for classes of problems with structure, such as network design, is explored.}, language = {en} } @misc{Helmberg1999, author = {Helmberg, Christoph}, title = {The m-Cost ATSP}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4374}, number = {SC-99-48}, year = {1999}, abstract = {Although the m-ATSP (or multi traveling salesman problem) is well known for its importance in scheduling and vehicle routing, it has, to the best of our knowledge, never been studied polyhedraly, i.e., it has always been transformed to the standard ATSP. This transformation is valid only if the cost of an arc from node \$i\$ to node \$j\$ is the same for all machines. In many practical applications this is not the case, machines produce with different speeds and require different (usually sequence dependent) setup times. We present first results of a polyhedral analysis of the m-ATSP in full generality. For this we exploit the tight relation between the subproblem for one machine and the prize collecting traveling salesman problem. We show that, for \$m\ge 3\$ machines, all facets of the one machine subproblem also define facets of the m-ATSP polytope. In particular the inequalities corresponding to the subtour elimination constraints in the one machine subproblems are facet defining for m-ATSP for \$m\ge 2\$ and can be separated in polynomial time. Furthermore, they imply the subtour elimination constraints for the ATSP-problem obtained via the standard transformation for identical machines. In addition, we identify a new class of facet defining inequalities of the one machine subproblem, that are also facet defining for m-ATSP for \$m\ge 2\$. To illustrate the efficacy of the approach we present numerical results for a scheduling problem with non-identical machines, arising in the production of gift wrap at Herlitz PBS AG.}, language = {en} } @misc{BorndoerferLoebelWeider2004, author = {Bornd{\"o}rfer, Ralf and L{\"o}bel, Andreas and Weider, Steffen}, title = {A Bundle Method for Integrated Multi-Depot Vehicle and Duty Scheduling in Public Transit}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7898}, number = {04-14}, year = {2004}, abstract = {This article proposes a Lagrangean relaxation approach to solve integrated duty and vehicle scheduling problems arising in public transport. The approach is based on the proximal bundle method for the solution of concave decomposable functions, which is adapted for the approximate evaluation of the vehicle and duty scheduling components. The primal and dual information generated by the bundle method is used to guide a branch-and-bound type algorithm. Computational results for large-scale real-world integrated vehicle and duty scheduling problems with up to 1,500 timetabled trips are reported. Compared with the results of a classical sequential approach and with reference solutions, integrated scheduling offers remarkable potentials in savings and drivers' satisfaction.}, language = {en} } @misc{Groetschel1999, author = {Gr{\"o}tschel, Martin}, title = {My Favorite Theorem: Characterizations of Perfect Graphs}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4053}, number = {SC-99-17}, year = {1999}, abstract = {This paper summarizes and discusses various characterizations of perfect graphs and mentions some open problems in this area.}, language = {en} } @misc{Koster2000, author = {Koster, Arie M.C.A.}, title = {Re-Optimization of Signaling Transfer Points}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5863}, number = {00-18}, year = {2000}, abstract = {In this paper we describe the results of a computational study towards the (re)optimization of signaling transfer points (STPs) in telecommunication networks. The best performance of an STP is achieved whenever the traffic load is evenly distributed among the internal components. Due to the continuously changing traffic pattern, the load of the components has to be re-optimized on a regular basis. Besides the balancing objective also the number of rearrangements have to be taken into account. In this paper we present two alternative formulations to deal with both requirements. Computational results show that for both formulations (near) optimal solutions can be obtained within reasonable time limits.}, language = {en} } @misc{Byun2001, author = {Byun, Chul-Young}, title = {Lower Bounds for Large-Scale Set Partitioning Problems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6321}, number = {01-06}, year = {2001}, abstract = {In this work we concentrate on developing methods which determine good lower bounds for set partitioning problems (SPP) in an appropriate amount of time. We found out that it makes sense to use the Lagrangian relaxation method for this task. The Lagrangian relaxed problem of SPP has a simple structure, which leads to algorithms and heuristics, whose total complexity per iteration depends linearly on the number of non-zeros of the problem matrix of SPP. In contrast, other methods like simplex methods or interior point methods have a complexity of higher order. Because the problem matrices of our tested instances are sparse, the linear dependence becomes an advantage for the algorithms and heuristics mentioned above. As a reference for the state-of-the-art we have applied the dual simplex method and the barrier function method, implemented in CPLEX. The methods, which we have developed and compared with those of CPLEX, are SBM, CAM, CCBM, and CBM. SBM is a subgradient bundle method derived from the basic subgradient method, which is a global convergent method for determining the maximum of concave functions. CAM is a coordinate ascent method, where the convex coordinate bundle method CCBM and the coordinate bundle method CBM are derivatives from CAM. We observed that the basic subgradient and the coordinate ascent method are improved if bundling techniques can be used. But the motivation for bundling differs for both approaches. In the former case bundling helps to approximate a minimum norm subgradient, which provides a steepest ascent direction, in order to speed up the performance. In the latter case bundling enables proceeding along directions, which are not restricted on the coordinate directions. By this the performance is accelerated. Among all used techniques stabilization is worth mentioning. Stabilization improves the performance especially at the beginning by avoiding too big steps during the proceeding. This leads to a more stabilized progression. Stabilization was successfully applied to SBM, CAM, CCBM, and CBM. As an overall result we conclude the following: \begin{enumerate} \item CPLEX computes the optimal objective values, whereas SBM and CBM has on average a gap of under \$1.5\\%\$. \item In comparison to CPLEX baropt, SBM, CAM, and CBM the algorithm CCBM has a slow convergence because of the convex combination of ascent coordinate directions. An alternative is to relax the convex combination to a simple sum of the corresponding directions. This idea is realized in CBM. \item If we focus on the running time rather than on optimality then CBM is on average the fastest algorithm. \end{enumerate} Note that methods like SBM or CBM are applied on static SPP instances in order to determine a good lower bound. For solving SPP we need dynamical methods. Due to the complex topic of dynamical methods we will not discuss them, but a certain technique is worth mentioning. It is called column generation. We have indicated that this technique needs good Lagrangian multipliers of the corresponding SPP instances in order to generate further columns (in our case duties), which are added to the current SPP instance. Those multipliers are by-products of methods like our six considered methods. Due to the large number of such generation steps the running time depends on the computation time of these methods. Therefore, CBM fits more to this technique than CPLEX baropt or SBM. To sum it up it can be said that applications such as a duty scheduling can be described as set partitioning problems, whose lower bound can be solved by simplex, interior points, subgradient, or coordinate ascent methods. It turns out that the interior points method CPLEX baropt and the heuristic CBM have good performances. Furthermore, good Lagrangian multipliers, which are by-products of these methods, can be used by techniques like column generation. For this particular technique it also turns out that among our tested algorithms CBM is the most efficient one. In general we can state that real-world applications, which have to solve a large number of Lagrangian relaxed SPP instances can improve their performance by using CBM.}, language = {en} } @misc{Wagler2002, author = {Wagler, Annegret}, title = {Relaxing Perfectness: Which Graphs are 'Almost' Perfect?}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6700}, number = {02-03}, year = {2002}, abstract = {For all perfect graphs, the stable set polytope STAB\$(G)\$ coincides with the fractional stable set polytope QSTAB\$(G)\$, whereas STAB\$(G) \subset\$ QSTAB\$(G)\$ holds iff \$G\$ is imperfect. Padberg asked in the early seventies for ``almost'' perfect graphs. He characterized those graphs for which the difference between STAB\$(G)\$ and QSTAB\$(G)\$ is smallest possible. We develop this idea further and define three polytopes between STAB\$(G)\$ and QSTAB\$(G)\$ by allowing certain sets of cutting planes only to cut off all the fractional vertices of QSTAB\$(G)\$. The difference between QSTAB\$(G)\$ and the largest of the three polytopes coinciding with STAB\$(G)\$ gives some information on the stage of imperfectness of the graph~\$G\$. We obtain a nested collection of three superclasses of perfect graphs and survey which graphs are known to belong to one of those three superclasses. This answers the question: which graphs are ``almost'' perfect?}, language = {en} } @misc{Wagler2002, author = {Wagler, Annegret}, title = {Antiwebs are Rank-Perfect}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6742}, number = {02-07}, year = {2002}, abstract = {\We discuss a nested collection of three superclasses of perfect graphs: near-perfect, rank-perfect, and weakly rank-perfect graphs. For that, we start with the description of the stable set polytope for perfect graphs and allow stepwise more general facets for the stable set polytopes of the graphs in each superclass. Membership in those three classes indicates how far a graph is away from being perfect. We investigate for webs and antiwebs to which of the three classes they belong. We provide a complete description of the facets of the stable set polytope for antiwebs (with help of a result due to Shepherd on near-bipartite graphs). The main result is that antiwebs are rankperfect.}, language = {en} } @misc{BorndoerferLoebelWeider2002, author = {Bornd{\"o}rfer, Ralf and L{\"o}bel, Andreas and Weider, Steffen}, title = {Integrierte Umlauf- und Dienstplanung im {\"O}ffentlichen Nahverkehr}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6773}, number = {02-10}, year = {2002}, abstract = {Wir beschreiben einen Ansatz zur integrierten Umlauf- und Dienstplanung im {\"o}ffentlichen Nahverkehr. Der Ansatz zielt auf die Verbesserung des Gesamtwirkungsgrades dieser beiden Planungsschritte und auf die besondere Planungsproblematik im Regionalverkehr. Wir entwickeln dazu mathematische Optimierungstechniken f{\"u}r den Einsatz in den Planungssystemen MICROBUS II und DIVA.}, language = {de} } @misc{BorndoerferScheltenSchlechteetal.2005, author = {Bornd{\"o}rfer, Ralf and Schelten, Uwe and Schlechte, Thomas and Weider, Steffen}, title = {A Column Generation Approach to Airline Crew Scheduling}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8713}, number = {05-37}, year = {2005}, abstract = {The airline crew scheduling problem deals with the construction of crew rotations in order to cover the flights of a given schedule at minimum cost. The problem involves complex rules for the legality and costs of individual pairings and base constraints for the availability of crews at home bases. A typical instance considers a planning horizon of one month and several thousand flights. We propose a column generation approach for solving airline crew scheduling problems that is based on a set partitioning model. We discuss algorithmic aspects such as the use of bundle techniques for the fast, approximate solution of linear programs, a pairing generator that combines Lagrangean shortest path and callback techniques, and a novel rapid branching'' IP heuristic. Computational results for a number of industrial instances are reported. Our approach has been implemented within the commercial crew scheduling system NetLine/Crew of Lufthansa Systems Berlin GmbH.}, language = {en} } @misc{KaibelKoch2006, author = {Kaibel, Volker and Koch, Thorsten}, title = {Mathematik f{\"u}r den Volkssport}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9225}, number = {06-28}, year = {2006}, abstract = {"`Volkssport Sudoku"' titelt der Stern in seiner Ausgabe vom 24. Mai2006. In der Tat traut sich derzeit kaum noch eine Zeitung, ohne Sudoku zu erscheinen. Die Begeisterung am L{\"o}sen dieser Zahlenr{\"a}tsel offenbart eine unvermutete Freude am algorithmischen Arbeiten. Mathematisch kann man Sudokus als lineare diophantische Gleichungssysteme mit Nichtnegativit{\"a}tsbedingungen formulieren. Solche ganzzahligen linearen Programme sind die wichtigsten Modellierungswerkzeuge in zahlreichen Anwendungsgebieten wie z.B. der Optimierung von Telekommunikations- und Verkehrsnetzen. Moderne Verfahren zur L{\"o}sung dieser Optimierungsprobleme sind durch Sudokus allerdings deutlich weniger zu beeindrucken als Zeitungsleser.}, language = {de} } @misc{KosterZymolkaKutschka2007, author = {Koster, Arie M.C.A. and Zymolka, Adrian and Kutschka, Manuel}, title = {Algorithms to Separate {0,1/2}-Chvatal-Gomory Cuts}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9531}, number = {07-10}, year = {2007}, abstract = {Chvatal-Gomory cuts are among the most well-known classes of cutting planes for general integer linear programs (ILPs). In case the constraint multipliers are either 0 or \$\frac{1}{2}\$, such cuts are known as \$\{0,\frac{1}{2}\}\$-cuts. It has been proven by Caprara and Fischetti (1996) that separation of \$\{0,\frac{1}{2}\}\$-cuts is NP-hard. In this paper, we study ways to separate \$\{0,\frac{1}{2}\}\$-cuts effectively in practice. We propose a range of preprocessing rules to reduce the size of the separation problem. The core of the preprocessing builds a Gaussian elimination-like procedure. To separate the most violated \$\{0,\frac{1}{2}\}\$-cut, we formulate the (reduced) problem as integer linear program. Some simple heuristic separation routines complete the algorithmic framework. Computational experiments on benchmark instances show that the combination of preprocessing with exact and/or heuristic separation is a very vital idea to generate strong generic cutting planes for integer linear programs and to reduce the overall computation times of state-of-the-art ILP-solvers.}, language = {en} } @misc{BorndoerferCardonha2009, author = {Bornd{\"o}rfer, Ralf and Cardonha, Carlos}, title = {A Binary Quadratic Programming Approach to the Vehicle Positioning Problem}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11242}, number = {09-12}, year = {2009}, abstract = {The Vehicle Positioning Problem (VPP) consists of the assignment of vehicles (buses, trams or trains) of a public transport or railway company to parking positions in a depot and to timetabled trips. Such companies have many different types of vehicles, and each trip can be performed only by vehicles of some of these types. These assignments are non-trivial due to the topology of depots. The parking positions are organized in tracks, which work as one- or two-sided stacks or queues. If a required type of vehicle is not available in the front of any track, shunting movements must be performed in order to change vehicles' positions, which is undesirable and should be avoided. In this text we present integer linear and non-linear programming formulations for some versions of the problem and compare them from a theoretical and a computational point of view.}, language = {en} } @misc{BorndoerferKarbstein2014, author = {Bornd{\"o}rfer, Ralf and Karbstein, Marika}, title = {Metric Inequalities for Routings on Direct Connections}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-44219}, year = {2014}, abstract = {We consider multi-commodity flow problems in which capacities are installed on paths. In this setting, it is often important to distinguish between flows on direct connection routes, using single paths, and flows that include path switching. We show that the concept of metric inequalities to characterize capacities that support a multi-commodity flow can be generalized to deal with direct connections.}, language = {en} }