LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Courbure discrète : théorie et applications

International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor

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

Advances and Novel Approaches in Discrete Optimization

Discrete optimization is an important area of Applied Mathematics with a broad spectrum of applications in many fields. This book results from a Special Issue in the journal Mathematics entitled ‘Advances and Novel Approaches in Discrete Optimization’. It contains 17 articles covering a broad spectrum of subjects which have been selected from 43 submitted papers after a thorough refereeing process. Among other topics, it includes seven articles dealing with scheduling problems, e.g., online scheduling, batching, dual and inverse scheduling problems, or uncertain scheduling problems. Other subjects are graphs and applications, evacuation planning, the max-cut problem, capacitated lot-sizing, and packing algorithms

Spatial development of the city of London in the later middle ages

The thesis is concerned with a particular spatial morphology - that of the City of London in the later Middle Ages - and the way that it evolved. It addresses itself to the organisation of space at the meso-scale, i.e. the way buildings were aggregated or arranged on the ground, and the external spaces that were formed. The work has two main aims: firstly, to discover the principles which governed the arrangement of buildings; and, secondly to develop a model which will simulate the actual development of building patterns in late medieval London. The work is divided into three main parts. Part I provides a general introduction. The overall form and development of the City are described, and the social and economic context considered by reference to secondary sources. An outline is given of some of the main social and economic forces operative in the late medieval period - c.1400 - c.1600 - and in the seventeenth century - the period from which most of the cartographic evidence is drawn. The principal sources - cartographic and documentary - are listed and discussed. Graph theory, the mathematical language used in this study, is briefly discussed, and some basic definitions given. Part II comprises the morphological analysis, the results of which are in all cases presented in verbal rather than in mathematical form. Firstly, the gross geometrical and topological properties of the urban pattern are described, and the connectivity between buildings analysed by reference to a graph-theoretic representation of a limited area of the City. Attention is then turned to the organisation of space within the house. A substantial body of plans, drawn from seventeenth-century property surveys, is analysed, using graph notation to describe the access patterns within the house. A typology of house plans is proposed on the basis of the access graphs, and a possible functional and social interpretation of the evidence is considered. Returning to the meso-scale, the urban structure is decomposed into its constituent elements. At the highest level is the block - a region surrounded by streets: at the lowest level is the individual building or unit. The block is divided into two zones - the perimeter zone and the interior zone - and each zone is subdivided into segments. It is argued that the segments correspond to historical property divisions which exercised a decisive effect on the evolution of building pattern. From a state description of the morphology, the analysis proceeds to a process description, based on historical evidence. The historical continuity of the property divisions is supported, and the historical process of segment development reconstructed, by reference to cartographic and documentary sources. From this analysis, the basic rules of building development are identified, and a concise description of the process of development is given for perimeter and interior segments. Finally, the process description is applied to various hypothetical block configurations. A computer simulation was used to examine two main properties: the access structure through the blocks, i.e. the way access is maintained (or not) from one segment to another: and the overall density of building development. The results obtained from the computer model are compared with the empirical evidence derived from plan analysis, and the similarities/differences discussed. The inference is drawn that the building pattern of late medieval London may be seen to result in large measure from the formal logic of the system, and can be accounted for on a probailistic basis. Part III summarises the results. The process description is seen as a form of shape grammar, and the relationship of this with other shape grammars is discussed. A programme of further work is outlined, in which it is hoped that the approach may be applied successfully to the analysis of other historical morphologies