Solving k-way Graph Partitioning Problems to Optimality: The Impact of Semidefinite Relaxations and the Bundle Method

This paper is concerned with computing global optimal solutions for maximum k-cut problems. We improve on the SBC algorithm of Ghaddar, Anjos and Liers in order to compute such solutions in less time. We extend the design principles of the successful BiqMac solver for maximum 2-cut to the general maximum k-cut problem. As part of this extension, we investigate different ways of choosing variables for branching.We also study the impact of the separation of clique inequalities within this new framework and observe that it frequently reduces the number of subproblems considerably. Our computational results suggest that the proposed approach achieves a drastic speedup in comparison to SBC, especially when k = 3. We also made a comparison with the orbitopal fixing approach of Kaibel, Peinhardt and Pfetsch. The results suggest that while their performance is better for sparse instances and larger values of k, our proposed approach is superior for smaller k and for dense instances of medium size. Furthermore, we used CPLEX for solving the ILP formulation underlying the orbitopal fixing algorithm and conclude that especially on dense instances the new algorithm outperforms CPLEX by far

Solving k-way Graph Partitioning Problems to Optimality: The Impact of Semidefinite Relaxations and the Bundle Method

This paper is concerned with computing global optimal solutions for maximum k-cut problems. We improve on the SBC algorithm of Ghaddar, Anjos and Liers in order to compute such solutions in less time. We extend the design principles of the successful BiqMac solver for maximum 2-cut to the general maximum k-cut problem. As part of this extension, we investigate different ways of choosing variables for branching.We also study the impact of the separation of clique inequalities within this new framework and observe that it frequently reduces the number of subproblems considerably. Our computational results suggest that the proposed approach achieves a drastic speedup in comparison to SBC, especially when k = 3. We also made a comparison with the orbitopal fixing approach of Kaibel, Peinhardt and Pfetsch. The results suggest that while their performance is better for sparse instances and larger values of k, our proposed approach is superior for smaller k and for dense instances of medium size. Furthermore, we used CPLEX for solving the ILP formulation underlying the orbitopal fixing algorithm and conclude that especially on dense instances the new algorithm outperforms CPLEX by far

Ganzzahlige und Gemischt-Ganzzahlige Reformulierungen Stochastischer, Ressourcenbeschränkter und Quadratischer Matching-Probleme

The matching problem is one of the intensely studied combinatorial optimization problems. Nevertheless, many real-world problems cannot be formulated as a pure matching problem. In this thesis we study four variants of the classical matching problem: The resource-constrained bipartite matching problem, a stochastic matching problem, a recoverable robust matching problem and the quadratic matching problem. The stochastic matching problem and the recoverable robust matching problem arise from an application to runway scheduling. This thesis investigates integer as well as mixed-integer reformulations of these four types of matching problems. It consists of two parts. In the first part we study an exact solution approach for the resource-constrained bipartite matching problem and the stochastic matching problem. It reformulates the integer programming formulation (IP) of these problems into a mixed-integer program (MIP) that uses few integer variables. To this end, affine TU decompositions of the constraint matrix play a major role. We derive several theoretical results arising from the decomposition of the constraint matrices of matching problems with resource constraints. These findings can be extended to the stochastic matching problem as the latter can be modeled as a bipartite matching problem with a special resource constraint. In a computational study we compare different MIP reformulations of resource-constrained and stochastic matching problems with their corresponding IP formulations. We show that in several settings, running times for solving instances to optimality can significantly be reduced, when the new MIP reformulations are used. Furthermore, we examine the complexity status of the recoverable robust matching problem. In a simplified version we can model it as a bipartite matching problem with a quadratic objective in the edge variables. In the second part we study the quadratic matching problem. It asks for a matching in a graph that optimizes a quadratic objective in the edge variables. In our solution approach, we strengthen the linearized IP formulation by cutting planes that are derived from facets of the corresponding matching problem where only one quadratic term occurs in the objective function. We present different reformulation techniques to strengthen these cutting planes. Based on these methods, we design and implement an exact branch-and-cut approach. We show that root bounds and running times for solving instances to optimality can be improved significantly, when the new approach is applied.Das Matching-Problem ist eines der am besten untersuchten kombinatorischen Optimierungsprobleme. Viele Fragestellungen aus der Praxis, können jedoch nicht als reines Matching-Problem formuliert werden. In dieser Arbeit untersuchen wir vier Varianten des klassischen Matching-Problems: Das ressourcenbeschränkte Matching-Problem, ein stochastisches Matching-Problem, ein robust-wiederherstellbares Matching-Problem und das quadratische Matching-Problem. Das stochastische und das robust-wiederherstellbare Matching-Problem finden im Bereich der Flugplanoptimierung Anwendung. Die Arbeit untersucht ganzzahlige sowie gemischt-ganzzahlige Reformulierungsansätze für die vier oben genannten Matching-Probleme. Sie besteht aus zwei Teilen. Im ersten Teil untersuchen wir eine exakte Lösungsmethode für das ressourcenbeschränkte und für das stochastische Matching-Problem. Die grundlegende Idee dabei ist, das ganzzahlige lineare Programm (IP) dieser beiden Probleme als ein gemischt-ganzzahliges Programm (MIP) mit nur wenigen ganzzahligen Variablen zu reformulieren. Dabei spielt das Konzept der affinen TU Dekomposition der Restriktionsmatrix eine zentrale Rolle. Wir leiten einige theoretischeResultate her, die sich bei der Dekomposition der Restriktionsmatrix von Matching-Problemen mit Ressourcenbeschränkung ergeben. Diese Ergebnisse lassen sich auf das stochastische Matching-Problem erweitern, da dieses als bipartites Matching-Problem mit einer speziellen Ressourcenbeschränkung modelliert werden kann. In einer Rechenstudie vergleichen wir verschiedene MIP Reformulierungen für ressourcenbeschränkte und stochastische Matching-Probleme mit den IP Formulierungen der beiden Probleme. Wir zeigen, dass in einigen Settings die Laufzeiten signifikant verbessert werden können, wenn die neuen MIP Reformulierungen verwendet werden. Das robust-wiederherstellbare Matching-Problem betrachten wir unter komplexitätstheoretischen Aspekten. In einer vereinfachten Version lässt sich dieses als bipartites Matching-Problem mit einer quadratischen Zielfunktion in den Kantenvariablen modellieren. Im zweiten Teil studieren wir das quadratische Matching-Problem. Beim quadratischen Matching-Problem ist ein Matching gesucht, das eine quadratische Zielfunktion in den Kantenvariablen optimiert. Die Idee unserer Lösungsmethode ist, die linearisierte IP Formulierung durch Schnittebenen zu verstärken, die Facetten für das Matching-Problem mit nur einem quadratischen Term in der Zielfunktion induzieren. Wir geben verschiedene Reformulierungsansätze an, um die hergeleiteten Schnittebenen zu verstärken. Basierend auf diesen Methoden implementieren wir ein Branch-and-Cut Verfahren. Wir zeigen, dass durch Anwendung der neuen verstärkten Schnittebenen die Wurzelschranken und dieLaufzeiten signifikant verbessert werden können

Guidelines for the use and interpretation of assays for monitoring autophagy

In 2008 we published the first set of guidelines for standardizing research in autophagy. Since then, research on this topic has continued to accelerate, and many new scientists have entered the field. Our knowledge base and relevant new technologies have also been expanding. Accordingly, it is important to update these guidelines for monitoring autophagy in different organisms. Various reviews have described the range of assays that have been used for this purpose. Nevertheless, there continues to be confusion regarding acceptable methods to measure autophagy, especially in multicellular eukaryotes. A key point that needs to be emphasized is that there is a difference between measurements that monitor the numbers or volume of autophagic elements (e.g., autophagosomes or autolysosomes) at any stage of the autophagic process vs. those that measure flux through the autophagy pathway (i.e., the complete process); thus, a block in macroautophagy that results in autophagosome accumulation needs to be differentiated from stimuli that result in increased autophagic activity, defined as increased autophagy induction coupled with increased delivery to, and degradation within, lysosomes (in most higher eukaryotes and some protists such as Dictyostelium) or the vacuole (in plants and fungi). In other words, it is especially important that investigators new to the field understand that the appearance of more autophagosomes does not necessarily equate with more autophagy. In fact, in many cases, autophagosomes accumulate because of a block in trafficking to lysosomes without a concomitant change in autophagosome biogenesis, whereas an increase in autolysosomes may reflect a reduction in degradative activity. Here, we present a set of guidelines for the selection and interpretation of methods for use by investigators who aim to examine macroautophagy and related processes, as well as for reviewers who need to provide realistic and reasonable critiques of papers that are focused on these processes. These guidelines are not meant to be a formulaic set of rules, because the appropriate assays depend in part on the question being asked and the system being used. In addition, we emphasize that no individual assay is guaranteed to be the most appropriate one in every situation, and we strongly recommend the use of multiple assays to monitor autophagy. In these guidelines, we consider these various methods of assessing autophagy and what information can, or cannot, be obtained from them. Finally, by discussing the merits and limits of particular autophagy assays, we hope to encourage technical innovation in the field

Guidelines for the use and interpretation of assays for monitoring autophagy

In 2008 we published the first set of guidelines for standardizing research in autophagy. Since then, research on this topic has continued to accelerate, and many new scientists have entered the field. Our knowledge base and relevant new technologies have also been expanding. Accordingly, it is important to update these guidelines for monitoring autophagy in different organisms. Various reviews have described the range of assays that have been used for this purpose. Nevertheless, there continues to be confusion regarding acceptable methods to measure autophagy, especially in multicellular eukaryotes. A key point that needs to be emphasized is that there is a difference between measurements that monitor the numbers or volume of autophagic elements (e.g., autophagosomes or autolysosomes) at any stage of the autophagic process vs. those that measure flux through the autophagy pathway (i.e., the complete process); thus, a block in macroautophagy that results in autophagosome accumulation needs to be differentiated from stimuli that result in increased autophagic activity, defined as increased autophagy induction coupled with increased delivery to, and degradation within, lysosomes (in most higher eukaryotes and some protists such as Dictyostelium) or the vacuole (in plants and fungi). In other words, it is especially important that investigators new to the field understand that the appearance of more autophagosomes does not necessarily equate with more autophagy. In fact, in many cases, autophagosomes accumulate because of a block in trafficking to lysosomes without a concomitant change in autophagosome biogenesis, whereas an increase in autolysosomes may reflect a reduction in degradative activity. Here, we present a set of guidelines for the selection and interpretation of methods for use by investigators who aim to examine macroautophagy and related processes, as well as for reviewers who need to provide realistic and reasonable critiques of papers that are focused on these processes. These guidelines are not meant to be a formulaic set of rules, because the appropriate assays depend in part on the question being asked and the system being used. In addition, we emphasize that no individual assay is guaranteed to be the most appropriate one in every situation, and we strongly recommend the use of multiple assays to monitor autophagy. In these guidelines, we consider these various methods of assessing autophagy and what information can, or cannot, be obtained from them. Finally, by discussing the merits and limits of particular autophagy assays, we hope to encourage technical innovation in the field

Guidelines for the use and interpretation of assays for monitoring autophagy

In 2008 we published the first set of guidelines for standardizing research in autophagy. Since then, research on this topic has continued to accelerate, and many new scientists have entered the field. Our knowledge base and relevant new technologies have also been expanding. Accordingly, it is important to update these guidelines for monitoring autophagy in different organisms. Various reviews have described the range of assays that have been used for this purpose. Nevertheless, there continues to be confusion regarding acceptable methods to measure autophagy, especially in multicellular eukaryotes. A key point that needs to be emphasized is that there is a difference between measurements that monitor the numbers or volume of autophagic elements (e.g., autophagosomes or autolysosomes) at any stage of the autophagic process vs. those that measure flux through the autophagy pathway (i.e., the complete process); thus, a block in macroautophagy that results in autophagosome accumulation needs to be differentiated from stimuli that result in increased autophagic activity, defined as increased autophagy induction coupled with increased delivery to, and degradation within, lysosomes (in most higher eukaryotes and some protists such as Dictyostelium) or the vacuole (in plants and fungi). In other words, it is especially important that investigators new to the field understand that the appearance of more autophagosomes does not necessarily equate with more autophagy. In fact, in many cases, autophagosomes accumulate because of a block in trafficking to lysosomes without a concomitant change in autophagosome biogenesis, whereas an increase in autolysosomes may reflect a reduction in degradative activity. Here, we present a set of guidelines for the selection and interpretation of methods for use by investigators who aim to examine macroautophagy and related processes, as well as for reviewers who need to provide realistic and reasonable critiques of papers that are focused on these processes. These guidelines are not meant to be a formulaic set of rules, because the appropriate assays depend in part on the question being asked and the system being used. In addition, we emphasize that no individual assay is guaranteed to be the most appropriate one in every situation, and we strongly recommend the use of multiple assays to monitor autophagy. In these guidelines, we consider these various methods of assessing autophagy and what information can, or cannot, be obtained from them. Finally, by discussing the merits and limits of particular autophagy assays, we hope to encourage technical innovation in the field