TY - GEN A1 - Ralphs, Ted A1 - Shinano, Yuji A1 - Berthold, Timo A1 - Koch, Thorsten T1 - Parallel Solvers for Mixed Integer Linear Programming N2 - In this article, we introduce parallel mixed integer linear programming (MILP) solvers. MILP solving algorithms have been improved tremendously in the last two decades. Currently, commercial MILP solvers are known as a strong optimization tool. Parallel MILP solver development has started in 1990s. However, since the improvements of solving algorithms have much impact to solve MILP problems than application of parallel computing, there were not many visible successes. With the spread of multi-core CPUs, current state-of-the-art MILP solvers have parallel implementations and researches to apply parallelism in the solving algorithm also getting popular. We summarize current existing parallel MILP solver architectures. T3 - ZIB-Report - 16-74 KW - parallel computing KW - integer programming KW - branch-and-bound KW - supercomputing Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-62239 SN - 1438-0064 ER - TY - GEN A1 - Achterberg, Tobias A1 - Bixby, Robert E. A1 - Gu, Zonghao A1 - Rothberg, Edward A1 - Weninger, Dieter T1 - Presolve Reductions in Mixed Integer Programming N2 - Mixed integer programming has become a very powerful tool for modeling and solving real-world planning and scheduling problems, with the breadth of applications appearing to be almost unlimited. A critical component in the solution of these mixed-integer programs is a set of routines commonly referred to as presolve. Presolve can be viewed as a collection of preprocessing techniques that reduce the size of and, more importantly, improve the ``strength'' of the given model formulation, that is, the degree to which the constraints of the formulation accurately describe the underlying polyhedron of integer-feasible solutions. As our computational results will show, presolve is a key factor in the speed with which we can solve mixed-integer programs, and is often the difference between a model being intractable and solvable, in some cases easily solvable. In this paper we describe the presolve functionality in the Gurobi commercial mixed-integer programming code. This includes an overview, or taxonomy of the different methods that are employed, as well as more-detailed descriptions of several of the techniques, with some of them appearing, to our knowledge, for the first time in the literature. T3 - ZIB-Report - 16-44 KW - integer programming KW - presolving KW - Gurobi Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-60370 SN - 1438-0064 ER - TY - GEN A1 - Leitner, Markus A1 - Ljubic, Ivana A1 - Sinnl, Markus A1 - Werner, Axel T1 - Two algorithms for solving 3-objective k-ArchConFL and IPs in general N2 - We present two algorithms to solve a 3-objective optimization problem arising in telecommunications access network planning, the k-Architecture Connected Facility Location Problem. The methods can also be used to solve any 3-objective integer linear programming model and can be extended to the multiobjective case. We give some exemplary computations using small and medium-sized instances for our problem. T3 - ZIB-Report - 15-48 KW - multiobjective optimization KW - integer programming KW - telecommunications Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-56259 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Karbstein, Marika T1 - Metric Inequalities for Routings on Direct Connections with Application in Line Planning N2 - 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 derive a feasibility condition for path capacities supporting such direct connection flows similar to the feasibility condition for arc capacities in ordinary multi-commodity flows. The concept allows to solve large-scale real-world line planning problems in public transport including a novel passenger routing model that favors direct connections over connections with transfers. T3 - ZIB-Report - 15-07 KW - combinatorial optimization KW - integer programming KW - line planning KW - transfers KW - direct connection KW - metric inequalities Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-53507 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Sagnol, Guillaume A1 - Schlechte, Thomas A1 - Swarat, Elmar T1 - Optimal Toll Enforcement - an Integration of Vehicle Routing and Duty Rostering N2 - We present the problem of planning mobile tours of inspectors on German motorways to enforce the payment of the toll for heavy good trucks. This is a special type of vehicle routing problem with the objective to conduct as good inspections as possible on the complete network. In addition, the crews of the tours have to be scheduled. Thus, we developed a personalized crew rostering model. The planning of daily tours and the rostering are combined in a novel integrated approach and formulated as a complex and large scale Integer Program. The paper focuses first on different requirements for the rostering and how they can be modeled in detail. The second focus is on a bicriterion analysis of the planning problem to find the balance between the control quality and the roster acceptance. On the one hand the tour planning is a profit maximization problem and on the other hand the rostering should be made in a employee friendly way. Finally, computational results on real-world instances show the practicability of our method. T3 - ZIB-Report - 13-79 KW - vehicle routing KW - crew rostering KW - integer programming KW - bicriteria optimization Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-45107 SN - 1438-0064 ER - TY - THES A1 - Raack, Christian T1 - Capacitated Network Design - Multi-Commodity Flow Formulations, Cutting Planes, and Demand Uncertainty N2 - In this thesis, we develop methods in mathematical optimization to dimension networks at minimal cost. Given hardware and cost models, the challenge is to provide network topologies and efficient capacity plans that meet the demand for network traffic (data, passengers, freight). We incorporate crucial aspects of practical interest such as the discrete structure of available capacities as well as the uncertainty of demand forecasts. The considered planning problems typically arise in the strategic design of telecommunication or public transport networks and also in logistics. One of the essential aspects studied in this work is the use of cutting planes to enhance solution approaches based on multi-commodity flow formulations. Providing theoretical and computational evidence for the efficacy of inequalities based on network cuts, we extend existing theory and algorithmic work in different directions. First, we prove that special-purpose techniques, originally designed to solve capacitated network design problems, can be successfully integrated into general-purpose mixed integer programming (MIP) solvers. Our approach relies on an automatic detection of network structure within the constraint matrix of general mixed in teger programs. More precisely, we identify multi-commodity (MCF) network sub-matrices and resolve the isomorphisms of the commodity blocks as well as the original graph structure. In the subsequent separation framework, we guide the constraint aggregation of available cutting plane procedures (e. g. based on mixed integer rounding) to produce strong cutting planes that reflect the structure of the constructed network. The new MCF-separator integrates network design specific methodology into general optimization tools which is of particular importance for practitioners that tend to use MIP solvers as black boxes. Extensive computational tests show that our network detection procedure operates accurately and reliably. Moreover, due to the generated cutting planes, we achieve an average speed-up of a factor of two for pure network design problems with general MIP solvers. Many of these instances can only be solved to optimality in reasonable time if the new MCF-separator is active. In 9 % of the instances of general MIP test sets we find consistent embedded networks and generate violated inequalities. In this case the computation time decreases by 18 % on average with almost no degradation for unaffected instances. Second, we generalize concepts, models, and cutting planes from deterministic network design to robust network design, incorporating the uncertainty of traffic demands. We enhance and compare strategies that are able to handle a polyhedral set of different traffic scenarios. In particular, we consider two correlated solution methods, based on separating extreme demand scenarios and dualizing the linear description of the demand polytope, respectively. We consider robust network design as two-stage robust optimization with recourse. First stage capacity decisions are fixed for all scenarios while the second stage flow depends on the realized demands. In order to reroute the traffic as a function of the demand dynamics, we consider three alternative recourse actions, namely, static, affine, and dynamic routing. We analyze properties of the new affine routing and show that it combines advantages of the well-known static and dynamic models. Using the concept of robust cut-set polyhedra and the corresponding lifting theorems, we develop several classes of facet-defining inequalities based on network cuts that can be used to further accelerate solution strategies for robust network design. Among them are the well-known (flow) cut-set inequalities, which we generalize to general demand polytopes, but also new classes of potential cutting planes, so-called envelope inequalities. The practical importance of the developed cutting planes is revealed by a series of computational tests. Similar to the results for the MCF-separator we achieve speed-ups of two and more using the generalized classes of strong inequalities. To evaluate the robustness of solutions that are computed with our framework we use real-life measurements of traffic dynamics from different existing telecommunication networks, among them data from the German and the European research network. Our results indicate that traffic peaks do not necessarily occur all simultaneously with respect to different source-destination pairs, which is of practical importance for the design of uncertainty sets. It is, in particular, not necessary to dimension networks for a scenario that assumes all source-destination traffic is at its peak simultaneously. With our solutions we save up to 20 % of the corresponding solution cost compared to this artificial scenario and achieve comparable levels of robustness. KW - combinatorial optimization KW - cutting planes KW - integer programming KW - network design KW - robustness Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:83-opus-36167 UR - http://opus4.kobv.de/opus4-tuberlin/frontdoor/index/index/docId/3407 ER - TY - GEN A1 - Raack, Christian A1 - Raymond, Annie A1 - Werner, Axel A1 - Schlechte, Thomas T1 - Integer Programming and Sports Rankings N2 - Sports rankings are obtained by applying a system of rules to evaluate the performance of the participants in a competition. We consider rankings that result from assigning an ordinal rank to each competitor according to their performance. We develop an integer programming model for rankings that allows us to calculate the number of points needed to guarantee a team the ith position, as well as the minimum number of points that could yield the ith place. The model is very general and can thus be applied to many types of sports. We discuss examples coming from football (soccer), ice hockey, and Formula~1. We answer various questions and debunk a few myths along the way. Are 40 points enough to avoid relegation in the Bundesliga? Do 95 points guarantee the participation of a team in the NHL playoffs? Moreover, in the season restructuration currently under consideration in the NHL, will it be easier or harder to access the playoffs? Is it possible to win the Formula~1 World Championship without winning at least one race or without even climbing once on the podium? Finally, we observe that the optimal solutions of the aforementioned model are associated to extreme situations which are unlikely to happen. Thus, to get closer to realistic scenarios, we enhance the model by adding some constraints inferred from the results of the previous years. T3 - ZIB-Report - 13-19 KW - sport ranking KW - integer programming Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-18068 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Karbstein, Marika ED - Delling, Daniel ED - Liberti, Leo T1 - A Direct Connection Approach to Integrated Line Planning and Passenger Routing T2 - ATMOS 2012 - 12th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems N2 - The treatment of transfers is a major challenge in line planning. Existing models either route passengers and lines sequentially, and hence disregard essential degrees of freedom, or they are of extremely large scale, and seem to be computationally intractable. We propose a novel direct connection approach that allows an integrated optimization of line and passenger routing, including accurate estimates of the number of direct travelers, for large-scale real-world instances. T3 - ZIB-Report - 12-29 KW - combinatorial optimization KW - integer programming KW - line planning KW - transfers Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-15812 SN - 1438-0064 VL - 25 SP - 47 EP - 57 PB - Schloss Dagstuhl - Leibniz-Zentrum für Informatik 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 - Borndörfer, Ralf A1 - Neumann, Marika T1 - Models for Line Planning with Transfers N2 - We propose a novel integer programming approach to transfer minimization for line planning problems in public transit. The idea is to incorporate penalties for transfers that are induced by “connection capacities” into the construction of the passenger paths. We show that such penalties can be dealt with by a combination of shortest and constrained shortest path algorithms such that the pricing problem for passenger paths can be solved efficiently. Connection capacity penalties (under)estimate the true transfer times. This error is, however, not a problem in practice. We show in a computational comparison with two standard models on a real-world scenario that our approach can be used to minimize passenger travel and transfer times for large-scale line planning problems with accurate results. T3 - ZIB-Report - 10-11 KW - Linienplanung KW - Umsteigen KW - Ganzzahlige Programmierung KW - Kombinatorische Optimierung KW - line planning KW - transfers KW - integer programming KW - combinatorial optimization Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11742 SN - 1438-0064 ER - TY - GEN A1 - Berthold, Timo A1 - Feydy, Thibaut A1 - Stuckey, Peter T1 - Rapid Learning for Binary Programs N2 - Learning during search allows solvers for discrete optimization problems to remember parts of the search that they have already performed and avoid revisiting redundant parts. Learning approaches pioneered by the SAT and CP communities have been successfully incorporated into the SCIP constraint integer programming platform. In this paper we show that performing a heuristic constraint programming search during root node processing of a binary program can rapidly learn useful nogoods, bound changes, primal solutions, and branching statistics that improve the remaining IP search. T3 - ZIB-Report - 10-04 KW - Ganzzahlige Programmierung KW - Constraintprogrammierung KW - Primalheuristik KW - Konfliktanalyse KW - constraint programming KW - integer programming KW - primal heuristic KW - conflict learning Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11663 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Cardonha, Carlos T1 - A Set Partitioning Approach to Shunting N2 - The Vehicle Positioning Problem (VPP) is a classical combinatorial optimization problem in public transport planning. A number of models and approaches have been suggested in the literature, which work for small problems, but not for large ones. We propose in this article a novel set partitioning model and an associated column generation solution approach for the VPP. The model provides a tight linear description of the problem. The pricing problem, and hence the LP relaxation itself, can be solved in polynomial resp. pseudo-polynomial time for some versions of the problems. T3 - ZIB-Report - 09-18 KW - integer programming KW - set partitioning KW - combinatorial packing KW - vehicle positioning Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11326 SN - 1438-0064 ER - TY - GEN A1 - Heinz, Stefan A1 - Stephan, Rüdiger A1 - Schlechte, Thomas T1 - Solving Steel Mill Slab Problems with Branch and Price N2 - The steel mill slab design problem from the CSPLib is a binpacking problem that is motivated by an application of the steel industry and that has been widely studied in the constraint programming community. Recently, several people proposed new models and methods to solve this problem. A steel mill slab library was created which contains 380 instances. A closely related binpacking problem called multiple knapsack problem with color constraints, originated from the same industrial problem, were discussed in the integer programming community. In particular, a simple integer programming for this problem has been given by Forrest et al. [3]. The aim of this paper is to bring these different studies together. Moreover, we adopt the model of [3] for the steel mill slab problem. Using a state of the art integer program solver, this model is capable to solve all instances of the steel mill slab library, mostly in less than one second, to optimality. We improved, thereby, the solution value of 76 instances. T3 - ZIB-Report - 09-14 KW - steel mill slab problem KW - branch-and-price KW - integer programming Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11260 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Grötschel, Martin A1 - Jaeger, Ulrich T1 - Planning Problems in Public Transit N2 - Every day, millions of people are transported by buses, trains, and airplanes in Germany. Public transit (PT) is of major importance for the quality of life of individuals as well as the productivity of entire regions. Quality and efficiency of PT systems depend on the political framework (state-run, market oriented) and the suitability of the infrastructure (railway tracks, airport locations), the existing level of service (timetable, flight schedule), the use of adequate technologies (information, control, and booking systems), and the best possible deployment of equipment and resources (energy, vehicles, crews). The decision, planning, and optimization problems arising in this context are often gigantic and “scream” for mathematical support because of their complexity. This article sketches the state and the relevance of mathematics in planning and operating public transit, describes today’s challenges, and suggests a number of innovative actions. The current contribution of mathematics to public transit is — depending on the transportation mode — of varying depth. Air traffic is already well supported by mathematics. Bus traffic made significant advances in recent years, while rail traffic still bears significant opportunities for improvements. In all areas of public transit, the existing potentials are far from being exhausted. For some PT problems, such as vehicle and crew scheduling in bus and air traffic, excellent mathematical tools are not only available, but used in many places. In other areas, such as rolling stock rostering in rail traffic, the performance of the existing mathematical algorithms is not yet sufficient. Some topics are essentially untouched from a mathematical point of view; e.g., there are (except for air traffic) no network design or fare planning models of practical relevance. PT infrastructure construction is essentially devoid of mathematics, even though enormous capital investments are made in this area. These problems lead to questions that can only be tackled by engineers, economists, politicians, and mathematicians in a joint effort. Among other things, the authors propose to investigate two specific topics, which can be addressed at short notice, are of fundamental importance not only for the area of traffic planning, should lead to a significant improvement in the collaboration of all involved parties, and, if successful, will be of real value for companies and customers: • discrete optimal control: real-time re-planning of traffic systems in case of disruptions, • model integration: service design in bus and rail traffic. Work on these topics in interdisciplinary research projects could be funded by the German ministry of research and education (BMBF), the German ministry of economics (BMWi), or the German science foundation (DFG). T3 - ZIB-Report - 09-13 KW - öffentlicher Verkehr KW - diskrete Optimierung KW - ganzzahlige Programmierung KW - public transit KW - discrete optimization KW - integer programming Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11252 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Cardonha, Carlos T1 - A Binary Quadratic Programming Approach to the Vehicle Positioning Problem N2 - 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. T3 - ZIB-Report - 09-12 KW - quadratic programming KW - integer programming KW - nonlinear programming KW - vehicle positioning Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11242 SN - 1438-0064 ER - TY - GEN A1 - Torres, Luis Miguel A1 - Torres, Ramiro A1 - Borndörfer, Ralf A1 - Pfetsch, Marc T1 - Line Planning on Paths and Tree Networks with Applications to the Quito Trolebus System (Extended Abstract) N2 - Line planning is an important step in the strategic planning process of a public transportation system. In this paper, we discuss an optimization model for this problem in order to minimize operation costs while guaranteeing a certain level of quality of service, in terms of available transport capacity. We analyze the problem for path and tree network topologies as well as several categories of line operation that are important for the Quito Trolebus system. It turns out that, from a computational complexity worst case point of view, the problem is hard in all but the most simple variants. In practice, however, instances based on real data from the Trolebus System in Quito can be solved quite well, and significant optimization potentials can be demonstrated. T3 - ZIB-Report - 08-53 KW - line planning KW - computational complexity KW - public transport KW - integer programming Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11076 SN - 1438-0064 ER - TY - GEN A1 - Torres, Luis Miguel A1 - Torres, Ramiro A1 - Borndörfer, Ralf A1 - Pfetsch, Marc T1 - Line Planning on Paths and Tree Networks with Applications to the Quito Trolebus System N2 - Line planning is an important step in the strategic planning process of a public transportation system. In this paper, we discuss an optimization model for this problem in order to minimize operation costs while guaranteeing a certain level of quality of service, in terms of available transport capacity. We analyze the problem for path and tree network topologies as well as several categories of line operation that are important for the Quito Trolebus system. It turns out that, from a computational complexity worst case point of view, the problem is hard in all but the most simple variants. In practice, however, instances based on real data from the Trolebus System in Quito can be solved quite well, and significant optimization potentials can be demonstrated. T3 - ZIB-Report - 08-35 KW - line planning KW - computational complexity KW - public transport KW - integer programming Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10869 SN - 1438-0064 ER - TY - GEN A1 - Berthold, Timo A1 - Pfetsch, Marc T1 - Detecting Orbitopal Symmetries N2 - Orbitopes can be used to handle symmetries which arise in integer programming formulations with an inherent assignment structure. We investigate the detection of symmetries appearing in this approach. We show that detecting so-called orbitopal symmetries is graph-isomorphism hard in general, but can be performed in linear time if the assignment structure is known. T3 - ZIB-Report - 08-33 KW - Symmetrie-Erkennung KW - Orbitope KW - Ganzzahlige Programmierung KW - Symmetrie-Brechung KW - Graphenisomorphie KW - symmetry detection KW - orbitopes KW - integer programming KW - symmetry breaking KW - graph ismorphism Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10842 SN - 1438-0064 ER - TY - GEN A1 - Berthold, Timo A1 - Heinz, Stefan A1 - Pfetsch, Marc T1 - Solving Pseudo-Boolean Problems with SCIP N2 - Pseudo-Boolean problems generalize SAT problems by allowing linear constraints and a linear objective function. Different solvers, mainly having their roots in the SAT domain, have been proposed and compared,for instance, in Pseudo-Boolean evaluations. One can also formulate Pseudo-Boolean models as integer programming models. That is,Pseudo-Boolean problems lie on the border between the SAT domain and the integer programming field. In this paper, we approach Pseudo-Boolean problems from the integer programming side. We introduce the framework SCIP that implements constraint integer programming techniques. It integrates methods from constraint programming, integer programming, and SAT-solving: the solution of linear programming relaxations, propagation of linear as well as nonlinear constraints, and conflict analysis. We argue that this approach is suitable for Pseudo-Boolean instances containing general linear constraints, while it is less efficient for pure SAT problems. We present extensive computational experiments on the test set used for the Pseudo-Boolean evaluation 2007. We show that our approach is very efficient for optimization instances and competitive for feasibility problems. For the nonlinear parts, we also investigate the influence of linear programming relaxations and propagation methods on the performance. It turns out that both techniques are helpful for obtaining an efficient solution method. T3 - ZIB-Report - 08-12 KW - Pseudo-Boolean KW - Constraint Programming KW - Ganzzahlige Programmierung KW - Branch-And-Cut KW - Optimierungssoftware KW - Pseudo-Boolean KW - constraint integer programming KW - integer programming KW - branch-and-cut KW - optimization software Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10671 SN - 1438-0064 ER - TY - GEN A1 - Achterberg, Tobias A1 - Heinz, Stefan A1 - Koch, Thorsten T1 - Counting solutions of integer programs using unrestricted subtree detection N2 - In the recent years there has been tremendous progress in the development of algorithms to find optimal solutions for integer programs. In many applications it is, however, desirable (or even necessary) to generate all feasible solutions. Examples arise in the areas of hardware and software verification and discrete geometry. In this paper, we investigate how to extend branch-and-cut integer programming frameworks to support the generation of all solutions. We propose a method to detect so-called unrestricted subtrees, which allows us to prune the integer program search tree and to collect several solutions simultaneously. We present computational results of this branch-and-count paradigm which show the potential of the unrestricted subtree detection. T3 - ZIB-Report - 08-09 KW - Zählen KW - ganzzahlige Programme KW - IP KW - counting KW - integer programming KW - IP Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10632 SN - 1438-0064 ER - TY - GEN A1 - Grötschel, Martin A1 - Nemhauser, George T1 - George Dantzig's contributions to integer programming N2 - This paper reviews George Dantzig's contribution to integer programming, especially his seminal work with Fulkerson and Johnson on the traveling salesman problem T3 - ZIB-Report - 07-39 KW - George Dantzig KW - integer programming KW - traveling salesman problem KW - TSP KW - mixed-integer programs Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10393 SN - 1438-0064 ER - TY - THES A1 - Achterberg, Tobias T1 - Constraint Integer Programming N2 - This thesis introduces the novel paradigm of constraint integer programming (CIP), which integrates constraint programming (CP) and mixed integer programming (MIP) modeling and solving techniques. It is supplemented by the software SCIP, which is a solver and framework for constraint integer programming that also features SAT solving techniques. SCIP is freely available in source code for academic and non-commercial purposes. Our constraint integer programming approach is a generalization of MIP that allows for the inclusion of arbitrary constraints, as long as they turn into linear constraints on the continuous variables after all integer variables have been fixed. The constraints, may they be linear or more complex, are treated by any combination of CP and MIP techniques: the propagation of the domains by constraint specific algorithms, the generation of a linear relaxation and its solving by LP methods, and the strengthening of the LP by cutting plane separation. The current version of SCIP comes with all of the necessary components to solve mixed integer programs. In the thesis, we cover most of these ingredients and present extensive computational results to compare different variants for the individual building blocks of a MIP solver. We focus on the algorithms and their impact on the overall performance of the solver. In addition to mixed integer programming, the thesis deals with chip design verification, which is an important topic of electronic design automation. Chip manufacturers have to make sure that the logic design of a circuit conforms to the specification of the chip. Otherwise, the chip would show an erroneous behavior that may cause failures in the device where it is employed. An important subproblem of chip design verification is the property checking problem, which is to verify whether a circuit satisfies a specified property. We show how this problem can be modeled as constraint integer program and provide a number of problem-specific algorithms that exploit the structure of the individual constraints and the circuit as a whole. Another set of extensive computational benchmarks compares our CIP approach to the current state-of-the-art SAT methodology and documents the success of our method. N2 - Diese Arbeit stellt einen integrierten Ansatz aus Constraint Programming (CP) und Gemischt-Ganzzahliger Programmierung (Mixed Integer Programming, MIP) vor, den wir Constraint Integer Programming (CIP) nennen. Sowohl Modellierungs- als auch Lösungstechniken beider Felder fließen in den neuen integrierten Ansatz ein, um die unterschiedlichen Stärken der beiden Gebiete zu kombinieren. Als weiteren Beitrag stellen wir der wissenschaftlichen Gemeinschaft die Software SCIP zur Verfügung, die ein Framework für Constraint Integer Programming darstellt und zusätzlich Techniken des SAT-Lösens beinhaltet. SCIP ist im Source Code für akademische und nicht-kommerzielle Zwecke frei erhältlich. Unser Ansatz des Constraint Integer Programming ist eine Verallgemeinerung von MIP, die zusätzlich die Verwendung beliebiger Constraints erlaubt, solange sich diese durch lineare Bedingungen ausdrücken lassen falls alle ganzzahligen Variablen auf feste Werte eingestellt sind. Die Constraints werden von einer beliebigen Kombination aus CP- und MIP-Techniken behandelt. Dies beinhaltet insbesondere die Domain Propagation, die Relaxierung der Constraints durch lineare Ungleichungen, sowie die Verstärkung der Relaxierung durch dynamisch generierte Schnittebenen. Die derzeitige Version von SCIP enthält alle Komponenten, die für das effiziente Lösen von Gemischt-Ganzzahligen Programmen benötigt werden. Die vorliegende Arbeit liefert eine ausführliche Beschreibung dieser Komponenten und bewertet verschiedene Varianten in Hinblick auf ihren Einfluß auf das Gesamt-Lösungsverhalten anhand von aufwendigen praktischen Experimenten. Dabei wird besonders auf die algorithmischen Aspekte eingegangen. Der zweite Hauptteil der Arbeit befasst sich mit der Chip-Design-Verifikation, die ein wichtiges Thema innerhalb des Fachgebiets der Electronic Design Automation darstellt. Chip-Hersteller müssen sicherstellen, dass der logische Entwurf einer Schaltung der gegebenen Spezifikation entspricht. Andernfalls würde der Chip fehlerhaftes Verhalten aufweisen, dass zu Fehlfunktionen innerhalb des Gerätes führen kann, in dem der Chip verwendet wird. Ein wichtiges Teilproblem in diesem Feld ist das Eigenschafts-Verifikations-Problem, bei dem geprüft wird, ob der gegebene Schaltkreisentwurf eine gewünschte Eigenschaft aufweist. Wir zeigen, wie dieses Problem als Constraint Integer Program modelliert werden kann und geben eine Reihe von problemspezifischen Algorithmen an, die die Struktur der einzelnen Constraints und der Gesamtschaltung ausnutzen. Testrechnungen auf Industrie-Beispielen vergleichen unseren Ansatz mit den bisher verwendeten SAT-Techniken und belegen den Erfolg unserer Methode. KW - Ganzzahlige Programmierung KW - Constraint Programmierung KW - SAT KW - Chip-Verifikation KW - integer programming KW - constraint programming KW - SAT KW - chip verification Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:83-opus-16117 ER -