TY - GEN A1 - Wolf, Thomas T1 - A Study of Genetic Algorithms solving a combinatorial Puzzle N2 - The suitability of Genetic Algorithms (GAs) to solve a combinatorial problem with only one solution is investigated. The dependence of the performance is studied for GA-hard and GA-soft fitness functions, both with a range of different parameter values and different encodings. T3 - ZIB-Report - SC-98-01 Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3445 ER - TY - GEN A1 - Harks, Tobias T1 - Nash Equilibria in Online Sequential Routing Games N2 - In this paper, we study the efficiency of Nash equilibria for a sequence of nonatomic routing games. We assume that the games are played consecutively in time in an online fashion: by the time of playing game $i$, future games $i+1,\dots,n$ are not known, and, once players of game $i$ are in equilibrium, their corresponding strategies and costs remain fixed. Given a sequence of games, the cost for the sequence of Nash equilibria is defined as the sum of the cost of each game. We analyze the efficiency of a sequence of Nash equilibria in terms of competitive analysis arising in the online optimization field. Our main result states that the online algorithm $\sl {SeqNash}$ consisting of the sequence of Nash equilibria is $\frac{4n}{2+n}$-competitive for affine linear latency functions. For $n=1$, this result contains the bound on the price of anarchy of $\frac{4}{3}$ for affine linear latency functions of Roughgarden and Tardos [2002] as a special case. Furthermore, we analyze a problem variant with a modified cost function that reflects the total congestion cost, when all games have been played. In this case, we prove an upper bound of $\frac{4n}{2+n}$ on the competitive ratio of $\sl {SeqNash}$. We further prove a lower bound of $\frac{3n-2}{n}$ of $\sl {SeqNash}$ showing that for $n=2$ our upper bound is tight. T3 - ZIB-Report - 06-43 KW - Congestion Game KW - Online Optimization KW - Nash equilibria Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9376 ER - TY - GEN A1 - Harks, Tobias A1 - Heinz, Stefan A1 - Pfetsch, Marc T1 - Competitive Online Multicommodity Routing N2 - We study online multicommodity minimum cost routing problems in networks, where commodities have to be routed sequentially. Arcs are equipped with load dependent price functions defining the routing weights. We discuss an online algorithm that routes each commodity by minimizing a convex cost function that depends on the demands that are previously routed. We present a competitive analysis of this algorithm showing that for affine linear price functions this algorithm is $4K/2+K$-competitive, where $K$ is the number of commodities. For the parallel arc case this algorithm is optimal. Without restrictions on the price functions and network, no algorithm is competitive. Finally, we investigate a variant in which the demands have to be routed unsplittably. T3 - ZIB-Report - 06-27 KW - Online Optimization KW - Routing KW - Telecommunications Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9212 ER - TY - GEN A1 - Harks, Tobias A1 - Heinz, Stefan A1 - Pfetsch, Marc T1 - Competitive Online Multicommodity Routing N2 - In this paper we study online multicommodity routing problems in networks, in which commodities have to be routed sequentially. The flow of each commodity can be split on several paths. Arcs are equipped with load dependent price functions defining routing costs, which have to be minimized. We discuss a greedy online algorithm that routes each commodity by minimizing a convex cost function that only depends on the demands previously routed. We present a competitive analysis of this algorithm showing that for affine linear price functions this algorithm is 4K2 (1+K)2 -competitive, where K is the number of commodities. For the single-source single-destination case, this algorithm is optimal. Without restrictions on the price functions and network, no algorithm is competitive. Finally, we investigate a variant in which the demands have to be routed unsplittably. T3 - ZIB-Report - 07-16 KW - Online Optimization KW - Routing KW - Telecommunications Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9599 ER - TY - THES A1 - Gamrath, Gerald T1 - Generic Branch-Cut-and-Price N2 - In this thesis, we present the theoretical background, implementational details and computational results concerning the generic branch-cut-and-price solver GCG. KW - mixed-integer programming KW - column generation KW - branch-and-price Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-57543 ER - TY - GEN A1 - Schiela, Anton A1 - Wachsmuth, Daniel T1 - Convergence Analysis of Smoothing Methods for Optimal Control of Stationary Variational Inequalities N2 - In the article an optimal control problem subject to a stationary variational inequality is investigated. The optimal control problem is complemented with pointwise control constraints. The convergence of a smoothing scheme is analyzed. There, the variational inequality is replaced by a semilinear elliptic equation. It is shown that solutions of the regularized optimal control problem converge to solutions of the original one. Passing to the limit in the optimality system of the regularized problem allows to prove C-stationarity of local solutions of the original problem. Moreover, convergence rates with respect to the regularization parameter for the error in the control are obtained. These rates coincide with rates obtained by numerical experiments, which are included in the paper. T3 - ZIB-Report - 11-23 KW - Variational inequalities KW - optimal control KW - control constraints KW - path-following KW - C-stationarity Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-13125 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Reuther, Markus A1 - Schlechte, Thomas A1 - Weider, Steffen T1 - A Hypergraph Model for Railway Vehicle Rotation Planning N2 - We propose a model for the integrated optimization of vehicle rotations and vehicle compositions in long distance railway passenger transport. The main contribution of the paper is a hypergraph model that is able to handle the challenging technical requirements as well as very general stipulations with respect to the ``regularity'' of a schedule. The hypergraph model directly generalizes network flow models, replacing arcs with hyperarcs. Although NP-hard in general, the model is computationally well-behaved in practice. High quality solutions can be produced in reasonable time using high performance Integer Programming techniques, in particular, column generation and rapid branching. We show that, in this way, large-scale real world instances of our cooperation partner DB Fernverkehr can be solved. T3 - ZIB-Report - 11-36 KW - Rolling Stock Planning, Hypergraph Modeling, Integer Programming, Column Generation, Rapid Branching Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0030-drops-32746 SN - 1438-0064 ER - TY - GEN A1 - Heinz, Stefan A1 - Schlechte, Thomas A1 - Stephan, Rüdiger A1 - Winkler, Michael T1 - Solving steel mill slab design problems N2 - The steel mill slab design problem from the CSPLIB is a combinatorial optimization problem motivated by an application of the steel industry. It has been widely studied in the constraint programming community. Several methods were proposed to solve this problem. A steel mill slab library was created which contains 380 instances. A closely related binpacking problem called the multiple knapsack problem with color constraints, originated from the same industrial problem, was discussed in the integer programming community. In particular, a simple integer program for this problem has been given by Forrest et al. The aim of this paper is to bring these different studies together. Moreover, we adapt the model of Forrest et al. for the steel mill slab design problem. Using this model and a state-of-the-art integer program solver all instances of the steel mill slab library can be solved efficiently to optimality. We improved, thereby, the solution values of 76 instances compared to previous results. Finally, we consider a recently introduced variant of the steel mill slab design problem, where within all solutions which minimize the leftover one is interested in a solution which requires a minimum number of slabs. For that variant we introduce two approaches and solve all instances of the steel mill slab library with this slightly changed objective function to optimality. T3 - ZIB-Report - 11-38 KW - steel mill slab design problem KW - multiple knapsack problem with color constraints KW - integer programming KW - set partitioning KW - binpacking with side constraints Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-14089 SN - 1438-0064 ER - TY - GEN A1 - Schlechte, Thomas T1 - Railway Track Allocation - Simulation and Optimization N2 - Today the railway timetabling process and the track allocation is one of the most challenging problems to solve by a railway infrastructure provider. Especially due to the deregulation of the transport market in the recent years several suppliers of railway traffic have entered the market. This leads to an increase of slot requests and then it is natural that conflicts occur among them. Furthermore, railway infrastructure networks consist of very expensive assets, even more they are rigid due to the long-term upgrade process. In order to make best use of these valuable infrastructure and to ensure economic operation, efficient planning of the railway operation is indispensable. Mathematical optimization models and algorithmic methodology can help to automatize and tackle these challenges. Our contribution in this paper is to present a renewed planning process due to the liberalization in Europe and a general framework to support the integration of simulation and optimization for railway capacity allocation. T3 - ZIB-Report - 11-32 KW - railway models KW - railway capacity KW - track allocation framework Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-13632 SN - 1438-0064 ER - TY - THES A1 - Hendel, Gregor T1 - New Rounding and Propagation Heuristics for Mixed Integer Programming N2 - Die vorliegende Arbeit befasst sich mit Primalheuristiken für gemischt-ganzzahlige, lineare Optimierungsprobleme (engl.: mixed integer program MIP). Zahlreiche Optimierungsprobleme aus der Praxis lassen sich als MIP modellieren, Beispiele hierfür sind u. a. Optimierungsprobleme im öffentlichen Nah- und Fernverkehr, bei logistischen Fragestellungen oder im Bereich der Chip-Verifikation. Das Lösen von MIP ist NP-schwer und wird heutzutage meistens mit Hilfe von Branch-and-Bound-basierenden Algorithmen versucht. Das Branch-and-Bound-Ver\-fah\-ren profitiert unter Umständen von bereits frühzeitig zur Verfügung stehenden Lösungen, daher sind wir sehr an heuristischen Verfahren interessiert, die in der Praxis schnell eine gute Lösung für eine große Zahl an MIPs liefern und somit die Lösezeit des Branch-and-Bound-Verfahrens erheblich beschleunigen können. Primalheuristiken sind Suchverfahren zum Auffinden zulässiger Lösungen eines MIP. Verschiedene Typen von Primalheuristiken sollen dabei den jeweiligen Bedarf des Anwenders zu unterschiedlichen Zeiten während der Branch-and-Bound-Suche decken. Während Start- und Rundeheuristiken zu Beginn des Löseprozesses eine große Rolle bei der Suche nach der ersten zulässigen Lösung haben, arbeiten Verbesserungs-heuristiken auf schon bekannten Lösungen, um neue, bessere Lösungen zu produzieren. Diese Arbeit beschäftigt sich mit Primalheuristiken, welche Teil des MIP-Lösers SCIP sind. Im ersten Kapitel werden nach der Erarbeitung grundlegender Definitionen viele der durch Tobias Achterberg und Timo Berthold in SCIP integrierten heuristischen Verfahren vorgestellt und kategorisiert. Auf dieser Grundlage bauen dann die Kapitel 2-4 der Arbeit auf. In diesen werden drei zusätzliche Heuristiken vorgestellt, im Einzelnen sind dies ZI Round, eine Rundeheuristik, welche zuerst von Wallace beschrieben wurde, außerdem eine 2-Opt-Heuristik für MIP und eine neue Startheuristik, Shift-And-Propagate. Großer Wert wird in jedem Kapitel auf die algorithmische Beschreibung der Heuristiken gelegt, die stets anhand von motivierenden Beispielen eingeführt und anhand von Pseudocode-Algorithmen begleitet werden. Zusätzlich enthält jedes Kapitel Auswertungen der mit den neuen Heuristiken gemessenen Ergebnisse von SCIP. Eine kurze Zusammenfassung in Kapitel 5 schließt diese Arbeit ab. N2 - Many practically relevant problems can be formulated in terms of a mixed integer programming (MIP) model. MIP denotes the optimization of a linear objective function under a certain number of linear side constraints including the need for some of the involved variables to take integral solution values. Applications of MIP based optimization can be found in the area of public transit, scheduling, automatic vehicle routing, network design, etc. From a complexity point of view, MIP solving is known to be NP-hard and most commonly tried to be solved via Branch-and-Bound based algorithms. Branch-and-Bound algorithms benefit from early and good feasible solutions of a MIP in various ways. Primal heuristics are aimed at finding new solutions during the MIP solving process. There are different types of primal heuristics: while start heuristics are particularly valuable to find an early solution, improvement heuristics hopefully drive a given solution further towards optimality. This thesis focusses on primal heuristics which are part of the MIP-solving framework SCIP. The first chapter comes with basic definitions and a brief description of SCIP and the test set which we used. The remainder of the first chapter is an overview of the existing heuristics in SCIP which have been implemented by Achterberg and Berthold. In the following chapters we introduce three new heuristics which apply rounding or propagation techniques for their specific purpose, namely the new rounding heuristic ZI Round, taken from Wallace, a 2-Opt improvement heuristic for MIP and the propagation heuristic Shift-and-Propagate. It is characteristic of all three heuristics that they mainly apply computationally inexpensive algorithms. Each of them is presented in an own chapter, starting with an algorithmic description, followed by implementational details. All chapters close with a discussion of the computational results obtained with the respective implementations in SCIP. KW - Primal Heuristics, MIP, mixed integer programming, ZI Round, Shift-and-Propagate, optimization Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-13322 ER - TY - GEN A1 - Berthold, Timo A1 - Heinz, Stefan A1 - Pfetsch, Marc A1 - Vigerske, Stefan T1 - Large Neighborhood Search beyond MIP N2 - Large neighborhood search (LNS) heuristics are an important component of modern branch-and-cut algorithms for solving mixed-integer linear programs (MIPs). Most of these LNS heuristics use the LP relaxation as the basis for their search, which is a reasonable choice in case of MIPs. However, for more general problem classes, the LP relaxation alone may not contain enough information about the original problem to find feasible solutions with these heuristics, e.g., if the problem is nonlinear or not all constraints are present in the current relaxation. In this paper, we discuss a generic way to extend LNS heuristics that have been developed for MIP to constraint integer programming (CIP), which is a generalization of MIP in the direction of constraint programming (CP). We present computational results of LNS heuristics for three problem classes: mixed-integer quadratically constrained programs, nonlinear pseudo-Boolean optimization instances, and resource-constrained project scheduling problems. Therefore, we have implemented extended versions of the following LNS heuristics in the constraint integer programming framework SCIP: Local Branching, RINS, RENS, Crossover, and DINS. Our results indicate that a generic generalization of LNS heuristics to CIP considerably improves the success rate of these heuristics. T3 - ZIB-Report - 11-21 KW - Large Neighborhood Search KW - Primal Heuristic KW - MIP KW - MIQCP KW - Pseudo-Boolean Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-12989 ER - TY - THES A1 - Schlechte, Thomas T1 - Railway Track Allocation: Models and Algorithms N2 - This thesis is about mathematical optimization for the efficient use of railway infrastructure. We address the optimal allocation of the available railway track capacity - the track allocation problem. This track allocation problem is a major challenge for a railway company, independent of whether a free market, a private monopoly, or a public monopoly is given. Planning and operating railway transportation systems is extremely hard due to the combinatorial complexity of the underlying discrete optimization problems, the technical intricacies, and the immense sizes of the problem instances. Mathematical models and optimization techniques can result in huge gains for both railway customers and operators, e.g., in terms of cost reductions or service quality improvements. We tackle this challenge by developing novel mathematical models and associated innovative algorithmic solution methods for large scale instances. This allows us to produce for the first time reliable solutions for a real world instance, i.e., the Simplon corridor in Switzerland. The opening chapter gives a comprehensive overview on railway planning problems. This provides insights into the regulatory and technical framework, it discusses the interaction of several planning steps, and identifies optimization potentials in railway transportation. The remainder of the thesis is comprised of two major parts. The first part is concerned with modeling railway systems to allow for resource and capacity analysis. Railway capacity has basically two dimensions, a space dimension which are the physical infrastructure elements as well as a time dimension that refers to the train movements, i.e., occupation or blocking times, on the physical infrastructure. Railway safety systems operate on the same principle all over the world. A train has to reserve infrastructure blocks for some time to pass through. Two trains reserving the same block of the infrastructure within the same point in time is called block conflict. Therefore, models for railway capacity involve the definition and calculation of reasonable running and associated reservation and blocking times to allow for a conflict free allocation. In the second and main part of the thesis, the optimal track allocation problem for macroscopic models of the railway system is considered. The literature for related problems is surveyed. A graph-theoretic model for the track allocation problem is developed. In that model optimal track allocations correspond to conflict-free paths in special time-expanded graphs. Furthermore, we made considerable progress on solving track allocation problems by two main features - a novel modeling approach for the macroscopic track allocation problem and algorithmic improvements based on the utilization of the bundle method. Finally, we go back to practice and present in the last chapter several case studies using the tools netcast and tsopt. We provide a computational comparison of our new models and standard packing models used in the literature. Our computational experience indicates that our approach, i.e., ``configuration models'', outperforms other models. Moreover, the rapid branching heuristic and the bundle method enable us to produce high quality solutions for very large scale instances, which has not been possible before. In addition, we present results for a theoretical and rather visionary auction framework for track allocation. We discuss several auction design questions and analyze experiments of various auction simulations. The highlights are results for the Simplon corridor in Switzerland. We optimized the train traffic through this tunnel using our models and software tools. To the best knowledge of the author and confirmed by several railway practitioners this was the first time that fully automatically produced track allocations on a macroscopic scale fulfill the requirements of the originating microscopic model, withstand the evaluation in the microscopic simulation tool OpenTrack, and exploit the infrastructure capacity. This documents the success of our approach in practice and the usefulness and applicability of mathematical optimization to railway track allocation. KW - railway track allocation KW - large-scale integer programming KW - network aggregation KW - rapid branching Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:83-opus-34272 UR - http://opus.kobv.de/tuberlin/volltexte/2012/3427/pdf/schlechte_thomas.pdf SN - 978-3-8381-3222-8 PB - Südwestdeutscher Verlag für Hochschulschriften CY - Saarbrücken, Germany ER - TY - GEN A1 - Reuther, Markus A1 - Borndörfer, Ralf A1 - Schlechte, Thomas A1 - Weider, Steffen T1 - Integrated Optimization of Rolling Stock Rotations for Intercity Railways N2 - This paper provides a highly integrated solution approach for rolling stock planning problems in the context of intercity passenger traffic. The main contributions are a generic hypergraph based mixed integer programming model and an integrated algorithm for the considered rolling stock rotation planning problem. The new developed approach is able to handle a very large set of industrial railway requirements, such as vehicle composition, maintenance constraints, infrastructure capacity, and regularity aspects. By the integration of this large bundle of technical railway aspects, we show that our approach has the power to produce implementable rolling stock rotations for our industrial cooperation partner DB Fernverkehr. This is the first time that the rolling stock rotations at DB Fernverkehr could be optimized by an automated system utilizing advanced mathematical programming techniques. T3 - ZIB-Report - 12-39 KW - Mixed Integer Programming KW - Railway Optimization KW - Rolling Stock Rostering Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-16424 SN - 1438-0064 ER - TY - GEN A1 - Hiller, Benjamin A1 - Vredeveld, Tjark T1 - Probabilistic alternatives for competitive analysis N2 - In the last 20 years competitive analysis has become the main tool for analyzing the quality of online algorithms. Despite of this, competitive analysis has also been criticized: It sometimes cannot discriminate between algorithms that exhibit significantly different empirical behavior, or it even favors an algorithm that is worse from an empirical point of view. Therefore, there have been several approaches to circumvent these drawbacks. In this survey, we discuss probabilistic alternatives for competitive analysis. T3 - ZIB-Report - 11-55 KW - online algorithms KW - probabilistic analysis KW - competitive analysis KW - survey Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-15131 SN - 1438-0064 ER - TY - GEN A1 - Schlechte, Thomas T1 - Railway Track Allocation N2 - This article gives an overview of the results of the author's PhD thesis. The thesis deals with the mathematical optimization for the efficient use of railway infrastructure. We address the optimal allocation of the available railway track capacity - the track allocation problem. This track allocation problem is a major challenge for a railway company, independent of whether a free market, a private monopoly, or a public monopoly is given. Planning and operating railway transportation systems is extremely hard due to the combinatorial complexity of the underlying discrete optimization problems, the technical intricacies, and the immense sizes of the problem instances. Mathematical models and optimization techniques can result in huge gains for both railway customers and operators, e.g., in terms of cost reductions or service quality improvements. We tackle this challenge by developing novel mathematical models and associated innovative algorithmic solution methods for large scale instances. We made considerable progress on solving track allocation problems by two main features - a novel modeling approach for the macroscopic track allocation problem and algorithmic improvements based on the utilization of the bundle method. This allows us to produce for the first time reliable solutions for a real world instance, i.e., the Simplon corridor in Switzerland. T3 - ZIB-Report - 12-38 KW - large-scale integer programming KW - network aggregation KW - configuration models KW - rapid branching Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-16402 SN - 1438-0064 ER - TY - GEN A1 - Sagnol, Guillaume T1 - Picos Documentation. Release 0.1.1. N2 - PICOS is a user friendly interface to several conic and integer programming solvers, very much like YALMIP under MATLAB. The main motivation for PICOS is to have the possibility to enter an optimization problem as a high level model, and to be able to solve it with several different solvers. Multidimensional and matrix variables are handled in a natural fashion, which makes it painless to formulate a SDP or a SOCP. This is very useful for educational purposes, and to quickly implement some models and test their validity on simple examples. Furthermore, with PICOS you can take advantage of the python programming language to read and write data, construct a list of constraints by using python list comprehensions, take slices of multidimensional variables, etc. T3 - ZIB-Report - 12-48 KW - mathematical programming KW - SDP KW - SOCP KW - python Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-17396 SN - 1438-0064 ER - TY - GEN A1 - Sagnol, Guillaume T1 - On the semidefinite representations of real functions applied to symmetric matrices N2 - We present a new semidefinite representation for the trace of a real function f applied to symmetric matrices, when a semidefinite representation of the convex function f is known. Our construction is intuitive, and yields a representation that is more compact than the previously known one. We also show with the help of matrix geometric means and the Riemannian metric of the set of positive definite matrices that for a rational number p in the interval (0,1], the matrix X raised to the exponent p is the largest element of a set represented by linear matrix inequalities. We give numerical results for a problem inspired from the theory of experimental designs, which show that the new semidefinite programming formulation yields a speed-up factor in the order of 10. T3 - ZIB-Report - 12-50 KW - semidefinite representability KW - optimal experimental designs KW - SDP KW - matrix geometric mean Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-17511 SN - 1438-0064 VL - 439 SP - 2829 EP - 2843 ER - TY - GEN A1 - Gamrath, Gerald T1 - Improving strong branching by propagation N2 - Strong branching is an important component of most variable selection rules in branch-and-bound based mixed-integer linear programming solvers. It predicts the dual bounds of potential child nodes by solving auxiliary LPs and thereby helps to keep the branch-and-bound tree small. In this paper, we describe how these dual bound predictions can be improved by including domain propagation into strong branching. Computational experiments on standard MIP instances indicate that this is beneficial in three aspects: It helps to reduce the average number of LP iterations per strong branching call, the number of branch-and-bound nodes, and the overall solving time. T3 - ZIB-Report - 12-46 KW - mixed-integer programming KW - branch-and-bound KW - branching rule KW - strong branching KW - domain propagation Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-17701 SN - 1438-0064 ER - TY - GEN A1 - Grötschel, Martin A1 - Stephan, Rüdiger T1 - Characterization of Facets of the Hop Constrained Chain Polytope via Dynamic Programming N2 - In this paper, we study the hop constrained chain polytope, that is, the convex hull of the incidence vectors of (s,t)-chains using at most k arcs of a given digraph, and its dominant. We use extended formulations (implied by the inherent structure of the Moore-Bellman-Ford algorithm) to derive facet defining inequalities for these polyhedra via projection. Our findings result into characterizations of all facet defining {0,+1,-1}-inequalities for the hop constrained chain polytope and all facet defining {0,1}-inequalities for its dominant. Although the derived inequalities are already known, such classifications were not previously given to the best of our knowledge. Moreover, we use this approach to generalize so called jump inequalities, which have been introduced in a paper of Dahl and Gouveia in 2004. T3 - ZIB-Report - 11-54 KW - hop constraints KW - chains KW - dynamic programming KW - facets Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-14914 SN - 1438-0064 ER - TY - GEN A1 - Sagnol, Guillaume T1 - Approximation of a Maximum-Submodular-Coverage problem involving spectral functions, with application to Experimental Design N2 - We study a family of combinatorial optimization problems defined by a parameter $p\in[0,1]$, which involves spectral functions applied to positive semidefinite matrices, and has some application in the theory of optimal experimental design. This family of problems tends to a generalization of the classical maximum coverage problem as $p$ goes to $0$, and to a trivial instance of the knapsack problem as $p$ goes to $1$. In this article, we establish a matrix inequality which shows that the objective function is submodular for all $p\in[0,1]$, from which it follows that the greedy approach, which has often been used for this problem, always gives a design within $1-1/e$ of the optimum. We next study the design found by rounding the solution of the continuous relaxed problem, an approach which has been applied by several authors. We prove an inequality which generalizes a classical result from the theory of optimal designs, and allows us to give a rounding procedure with an approximation factor which tends to $1$ as $p$ goes to $1$. T3 - ZIB-Report - 11-53 KW - Maximum Coverage KW - Optimal Experimental Design KW - Polynomial Time Approximability KW - Kiefer's p-criterion Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-14956 SN - 1438-0064 VL - 151 IS - 1--2 SP - 258 EP - 276 ER -