Exact Approaches for Higher-Dimensional Orthogonal Packing and Related Problems

NP-hard problems of higher-dimensional orthogonal packing are considered. We look closer at their logical structure and show that they can be decomposed into problems of a smaller dimension with a special contiguous structure. This decomposition influences the modeling of the packing process, which results in three new solution approaches. Keeping this decomposition in mind, we model the smaller-dimensional problems in a single position-indexed formulation with non-overlapping inequalities serving as binding constraints. Thus, we come up with a new integer linear programming model, which we subject to polyhedral analysis. Furthermore, we establish general non-overlapping and density inequalities and prove under appropriate assumptions their facet-defining property for the convex hull of the integer solutions. Based on the proposed model and the strong inequalities, we develop a new branch-and-cut algorithm. Being a relaxation of the higher-dimensional problem, each of the smaller-dimensional problems is also relevant for different areas, e.g. for scheduling. To tackle any of these smaller-dimensional problems, we use a Gilmore-Gomory model, which is a Dantzig-Wolfe decomposition of the position-indexed formulation. In order to obtain a contiguous structure for the optimal solution, its basis matrix must have a consecutive 1's property. For construction of such matrices, we develop new branch-and-price algorithms which are distinguished by various strategies for the enumeration of partial solutions. We also prove some characteristics of partial solutions, which tighten the slave problem of column generation. For a nonlinear modeling of the higher-dimensional packing problems, we investigate state-of-the-art constraint programming approaches, modify them, and propose new dichotomy and intersection branching strategies. To tighten the constraint propagation, we introduce new pruning rules. For that, we apply 1D relaxation with intervals and forbidden pairs, an advanced bar relaxation, 2D slice relaxation, and 1D slice-bar relaxation with forbidden pairs. The new rules are based on the relaxation by the smaller-dimensional problems which, in turn, are replaced by a linear programming relaxation of the Gilmore-Gomory model. We conclude with a discussion of implementation issues and numerical studies of all proposed approaches.Es werden NP-schwere höherdimensionale orthogonale Packungsprobleme betrachtet. Wir untersuchen ihre logische Struktur genauer und zeigen, dass sie sich in Probleme kleinerer Dimension mit einer speziellen Nachbarschaftsstruktur zerlegen lassen. Dies beeinflusst die Modellierung des Packungsprozesses, die ihreseits zu drei neuen Lösungsansätzen führt. Unter Beachtung dieser Zerlegung modellieren wir die Probleme kleinerer Dimension in einer einzigen positionsindizierten Formulierung mit Nichtüberlappungsungleichungen, die als Bindungsbedingungen dienen. Damit entwickeln wir ein neues Modell der ganzzahligen linearen Optimierung und unterziehen dies einer Polyederanalyse. Weiterhin geben wir allgemeine Nichtüberlappungs- und Dichtheitsungleichungen an und beweisen unter geeigneten Annahmen ihre facettendefinierende Eigenschaft für die konvexe Hülle der ganzzahligen Lösungen. Basierend auf dem vorgeschlagenen Modell und den starken Ungleichungen entwickeln wir einen neuen Branch-and-Cut-Algorithmus. Jedes Problem kleinerer Dimension ist eine Relaxation des höherdimensionalen Problems. Darüber hinaus besitzt es Anwendungen in verschiedenen Bereichen, wie zum Beispiel im Scheduling. Für die Behandlung der Probleme kleinerer Dimension setzen wir das Gilmore-Gomory-Modell ein, das eine Dantzig-Wolfe-Dekomposition der positionsindizierten Formulierung ist. Um eine Nachbarschaftsstruktur zu erhalten, muss die Basismatrix der optimalen Lösung die consecutive-1’s-Eigenschaft erfüllen. Für die Konstruktion solcher Matrizen entwickeln wir neue Branch-and-Price-Algorithmen, die sich durch Strategien zur Enumeration von partiellen Lösungen unterscheiden. Wir beweisen auch einige Charakteristiken von partiellen Lösungen, die das Hilfsproblem der Spaltengenerierung verschärfen. Für die nichtlineare Modellierung der höherdimensionalen Packungsprobleme untersuchen wir moderne Ansätze des Constraint Programming, modifizieren diese und schlagen neue Dichotomie- und Überschneidungsstrategien für die Verzweigung vor. Für die Verstärkung der Constraint Propagation stellen wir neue Ablehnungskriterien vor. Wir nutzen dabei 1D Relaxationen mit Intervallen und verbotenen Paaren, erweiterte Streifen-Relaxation, 2D Scheiben-Relaxation und 1D Scheiben-Streifen-Relaxation mit verbotenen Paaren. Alle vorgestellten Kriterien basieren auf Relaxationen durch Probleme kleinerer Dimension, die wir weiter durch die LP-Relaxation des Gilmore-Gomory-Modells abschwächen. Wir schließen mit Umsetzungsfragen und numerischen Experimenten aller vorgeschlagenen Ansätze

Algorithms for Highly Symmetric Linear and Integer Programs

This paper deals with exploiting symmetry for solving linear and integer programming problems. Basic properties of linear representations of finite groups can be used to reduce symmetric linear programming to solving linear programs of lower dimension. Combining this approach with knowledge of the geometry of feasible integer solutions yields an algorithm for solving highly symmetric integer linear programs which only takes time which is linear in the number of constraints and quadratic in the dimension.Comment: 21 pages, 1 figure; some references and further comments added, title slightly change

An exact approach for the vehicle routing problem with two-dimensional loading constraints

We consider a special case of the symmetric capacitated vehicle routing problem, in which a fleet of K identical vehicles must serve n customers, each with a given demand consisting in a set of rectangular two-dimensional weighted items. The vehicles have a two-dimensional loading surface and a maximum weight capacity. The aim is to find a partition of the customers into routes of minimum total cost such that, for each vehicle, the weight capacity is taken into account and a feasible two-Dimensional allocation of the items into the loading surface exists. The problem has several practical applications in freight transportation, and it is -hard in the strong sense. We propose an exact approach, based on a branch-and-cut algorithm, for the minimization of the routing cost that iteratively calls a branch-and-bound algorithm for checking the feasibility of the loadings. Heuristics are also used to improve the overall performance of the algorithm. The effectiveness of the approach is shown by means of computational results

Logic based Benders' decomposition for orthogonal stock cutting problems

We consider the problem of packing a set of rectangular items into a strip of fixed width, without overlapping, using minimum height. Items must be packed with their edges parallel to those of the strip, but rotation by 90\ub0 is allowed. The problem is usually solved through branch-and-bound algorithms. We propose an alternative method, based on Benders' decomposition. The master problem is solved through a new ILP model based on the arc flow formulation, while constraint programming is used to solve the slave problem. The resulting method is hybridized with a state-of-the-art branch-and-bound algorithm. Computational experiments on classical benchmarks from the literature show the effectiveness of the proposed approach. We additionally show that the algorithm can be successfully used to solve relevant related problems, like rectangle packing and pallet loading

Scheduling Kernels via Configuration LP

Makespan minimization (on parallel identical or unrelated machines) is arguably the most natural and studied scheduling problem. A common approach in practical algorithm design is to reduce the size of a given instance by a fast preprocessing step while being able to recover key information even after this reduction. This notion is formally studied as kernelization (or simply, kernel) - a polynomial time procedure which yields an equivalent instance whose size is bounded in terms of some given parameter. It follows from known results that makespan minimization parameterized by the longest job processing time p_max has a kernelization yielding a reduced instance whose size is exponential in p_max. Can this be reduced to polynomial in p_max? We answer this affirmatively not only for makespan minimization, but also for the (more complicated) objective of minimizing the weighted sum of completion times, also in the setting of unrelated machines when the number of machine kinds is a parameter. Our algorithm first solves the Configuration LP and based on its solution constructs a solution of an intermediate problem, called huge N-fold integer programming. This solution is further reduced in size by a series of steps, until its encoding length is polynomial in the parameters. Then, we show that huge N-fold IP is in NP, which implies that there is a polynomial reduction back to our scheduling problem, yielding a kernel. Our technique is highly novel in the context of kernelization, and our structural theorem about the Configuration LP is of independent interest. Moreover, we show a polynomial kernel for huge N-fold IP conditional on whether the so-called separation subproblem can be solved in polynomial time. Considering that integer programming does not admit polynomial kernels except for quite restricted cases, our "conditional kernel" provides new insight

Algorithms for highly symmetric linear and integer programs

This paper deals with exploiting symmetry for solving linear and integer programming problems. Basic properties of linear representations of finite groups can be used to reduce symmetric linear programming to solving linear programs of lower dimension. Combining this approach with knowledge of the geometry of feasible integer solutions yields an algorithm for solving highly symmetric integer linear programs which only takes time which is linear in the number of constraints and quadratic in the dimensio

Algorithms for highly symmetric linear and integer programs

Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch)This paper deals with exploiting symmetry for solving linear and integer programming problems. Basic properties of linear representations of finite groups can be used to reduce symmetric linear programming to solving linear programs of lower dimension. Combining this approach with knowledge of the geometry of feasible integer solutions yields an algorithm for solving highly symmetric integer linear programs which only takes time which is linear in the number of constraints and quadratic in the dimension