G. Mathematics of Computing
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The covering of a graph with (possibly disjoint) connected subgraphs is a fundamental problem in graph theory. In this paper, we study a version to cover a graph's vertices by connected subgraphs subject to lower and upper weight bounds, and propose a column generation approach to dynamically generate feasible and promising subgraphs. Our focus is on the solution of the pricing problem which turns out to be a variant of the NP-hard Maximum Weight Connected Subgraph Problem. We compare different formulations to handle connectivity, and find that a single-commodity flow formulation performs best. This is notable since the respective literature seems to have dismissed this formulation. We improve it to a new coarse-to-fine flow formulation that is theoretically and computationally superior, especially for large instances with many vertices of degree 2 like highway networks, where it provides a speed-up factor of 10 over the non-flow-based formulations. We also propose a preprocessing method that exploits a median property of weight constrained subgraphs, a primal heuristic, and a local search heuristic. In an extensive computational study we evaluate the presented connectivity formulations on different classes of instances, and demonstrate the effectiveness of the proposed enhancements. Their speed-ups essentially multiply to an overall factor of 20. Overall, our approach allows the reliabe solution of instances with several hundreds of nodes in a few minutes. These findings are further corroborated in a comparison to existing districting models on a set of test instances from the literature.
We investigate polyhedral aspects of the Periodic Event Scheduling Problem (PESP), the mathematical basis for periodic timetabling problems in public transport. Flipping the orientation of arcs, we obtain a new class of valid inequalities, the flip inequalities, comprising both the known cycle and change-cycle inequalities. For a point of the LP relaxation, a violated flip inequality can be found in pseudo-polynomial time, and even in linear time for a spanning tree solution. Our main result is that the integer vertices of the polytope described by the flip inequalities are exactly the vertices of the PESP polytope, i.e., the convex hull of all feasible periodic slacks with corresponding modulo parameters. Moreover, we show that this flip polytope equals the PESP polytope in some special cases. On the computational side, we devise several heuristic approaches concerning the separation of cutting planes from flip inequalities. These produce better dual bounds for the smallest and largest instance of the benchmarking library PESPlib.
The most important ingredient for solving mixed-integer nonlinear programs (MINLPs) to global epsilon-optimality with spatial branch and bound is a tight, computationally tractable relaxation. Due to both theoretical and practical considerations, relaxations of MINLPs are usually required to be convex. Nonetheless, current optimization solver can often successfully handle a moderate presence of nonconvexities, which opens the door for the use of potentially tighter nonconvex relaxations. In this work, we exploit this fact and make use of a nonconvex relaxation obtained via aggregation of constraints: a surrogate relaxation. These relaxations were actively studied for linear integer programs in the 70s and 80s, but they have been scarcely considered since. We revisit these relaxations in an MINLP setting and show the computational benefits and challenges they can have. Additionally, we study a generalization of such relaxation that allows for multiple aggregations simultaneously and present the first algorithm that is capable of computing the best set of aggregations. We propose a multitude of computational enhancements for improving its practical performance and evaluate the algorithm’s ability to generate strong dual bounds through extensive computational experiments.
In the planning process of public transportation companies, designing the timetable is among the core planning steps. In particular in the case of periodic (or cyclic) services, the Periodic Event Scheduling Problem (PESP) is well-established to compute high-quality periodic timetables.
We are considering algorithms for computing good solutions for the very basic PESP with no additional extra features as add-ons. The first of these algorithms generalizes several primal heuristics that had been proposed in the past, such as single-node cuts and the modulo network simplex algorithm. We consider partitions of the graph, and identify so-called delay cuts as a structure that allows to generalize several previous heuristics. In particular, when no more improving delay cut can be found, we already know that the other heuristics could not improve either.
The second of these algorithms turns a strategy, that had been discussed in the past, upside-down: Instead of gluing together the network line-by-line in a bottom-up way, we develop a divide-and-conquer-like top-down approach to separate the initial problem into two easier subproblems such that the information loss along their cutset edges is as small as possible.
We are aware that there may be PESP instances that do not fit well the separator setting. Yet, on the RxLy-instances of PESPlib in our experimental computations, we come up with good primal solutions and dual bounds. In particular, on the largest instance (R4L4), this new separator approach, which applies a state-of-the-art solver as subroutine, is able to come up with better dual bounds than purely applying this state-of-the-art solver in the very same time.
One of the most fundamental ingredients in mixed-integer nonlinear programming solvers is the well- known McCormick relaxation for a product of two variables x and y over a box-constrained domain. The starting point of this paper is the fact that the convex hull of the graph of xy can be much tighter when computed over a strict, non-rectangular subset of the box. In order to exploit this in practice, we propose to compute valid linear inequalities for the projection of the feasible region onto the x-y-space by solving a sequence of linear programs akin to optimization-based bound tightening. These valid inequalities allow us to employ results from the literature to strengthen the classical McCormick relaxation. As a consequence, we obtain a stronger convexification procedure that exploits problem structure and can benefit from supplementary information obtained during the branch-and bound algorithm such as an objective cutoff. We complement this by a new bound tightening procedure that efficiently computes the best possible bounds for x, y, and xy over the available projections. Our computational evaluation using the academic solver SCIP exhibit that the proposed methods are applicable to a large portion of the public test library MINLPLib and help to improve performance significantly.
Das Thema dieser Arbeit ist ein Volumen-Algorithmus für die Vereinigung von Polytopen. Der Algorithmus basiert auf der Arbeit von Bieri und Nef. Er berechnet das Volumen der Vereinigung von Polytopen mit einem Sweep-Verfahren. Dabei wird eine Hyperebene im Raum verschoben und das Volumen auf der einen Seite der Hyperebene berechnet. Umso weiter die Hyperebene verschobe wird, desto größer ist auch der Halbraum. Unser Algorithmus berechnet das Volumen einer Vereinigung von Polytopen geschnitten mit dem Halbraum der Sweep-Ebene als eine Funktion abhängig von der Veschiebung. Ab einem gewissen Punkt liegt der Körper dabei komplett im Halbraum der Sweep-Ebene und das Volumen bleibt konstant.
Unser Algorithmus unterscheidet sich in zwei Punkten von dem Algorithmus von Bieri und Nef. Erstens funktioniert er nur auf der Vereinigung von Polytopen, wohingegen der Algorithmus von Bieri und Nef für Nef-Polyeder funktioniert. Diese sind eine Verallgemeinerung von Polyedern, die auch die Klasse der Vereinigung von Polytopen umfasst. Für uns ist das allerdings kein Nachteil, da unsere Datensätze zu Vereinigungen von Polytopen führen. Zweitens ist unser Algorithmus in zwei Teile aufgeteilt. Im ersten Teil wird eine Datenstruktur entwickelt, aus der im zweiten Teil zusammen mit einer Richtung die Sweep-Ebenen-Volumenfunktion berechnet wird. Der Großteil der Komplexität liegt im ersten Teil des Algorithmus. Das hat den Vorteil, dass wir die Volumenfunktionen für viele verschiedene Richtungen berechnen können. So können Einblicke in die Struktur des Körpers gewonnen werden.
Der Algorithmus beruht auf zwei verschiedenen Zerlegungsansätzen. Zuerst können wir mit Hilfe von Anordnungen von Hyperebenen eine Vereinigung von Polytopen in ihre Zellen zerlegen. Dabei berufen wir uns auf die Arbeit von Gerstner und Holtz, in der das Konzept der Positionsvektoren eingeführt wird. Diese nutzen wir um die Ecken und ihre benachbarten Zellen zu bestimmen. So erhalten wir eine Zerlegung unserer Vereinigung in Zellen, deren paarweise Schnitte kein Volumen haben. Das zweite Zerlegungskonzept ist die konische Zerlegung, wie sie von Lawrence eingeführt wurde. Mit Hilfe dieser können wir die Indikatorfunktion eines Polytops als die Summe der Indikatorfunktionen seiner Vorwärtskegel schreiben. Die Sweep-Ebenen Volumenfunktionen können dann leicht mit Hilfe einer altbekannten Formel für das Volumen von Simplices berechnet werden.
During the past years hospitals saw themselves confronted with increasing economical pressure (WB06, p. V). Therefore, optimizing the general operational procedures has gained in importance. The revenue of a hospital depends on the kinds and quantity of treatments performed and on the effcient use and utilization of the corresponding resources. About 25 − 50% of the treatment costs of a patient needing surgery incurs in the operating rooms (WB06, p. 58). Hence skillful management of the operating rooms can have a large impact on the overall revenue of a hospital. Belien and Demeulemeester (BD07) describe the planning of operating room (OR) schedules as a multi-stage process. In the first stage OR time is allocated to the hospitals specialties and capacities and resources are adjusted. In the second stage a master surgery schedule (MSS) is developed, that is a timetable for D days that specifies the amount of OR time assigned to the specialties on every individual day. After D days this schedule will be repeated without any changes. Hence, developing an MSS is a long-term problem. Finally, specialties will schedule specific surgeries within their assigned OR time. In this work we will focus on the development of the MSS that maximizes the revenue of the hospital. Our main focus will be to ensure that the capacities of the downstream resources, i.e. the bed capacities in the ICU and ward, will not be exceeded. Additionally, we hope that our formulation of the problem will lead to a leveled bed demand without significant peaks. We will incorporate the uncertainty of patient demand and case mix in our model. There have been several approaches on this subject, for example in (Fü15) and (BD07) and this work is in part inuenced by these advances.
Improving relaxations for potential-driven network flow problems via acyclic flow orientations
(2018)
The class of potential-driven network flow problems provides important models for a range of infrastructure networks. For real-world applications, they need to be combined with integer
models for switching certain network elements, giving rise to hard-to-solve MINLPs. We observe that on large-scale real-world meshed networks the usually employed relaxations are rather weak due to cycles in the network.
We propose acyclic flow orientations as a combinatorial relaxation of feasible solutions of potential-driven flow problems and show how they can be used to strengthen existing relaxations. First computational results indicate that the strengthend model is much tighter than the original relaxation, thus promising a computational advantage.