Combinatorics and Geometry of Transportation Polytopes: An Update

A transportation polytope consists of all multidimensional arrays or tables of non-negative real numbers that satisfy certain sum conditions on subsets of the entries. They arise naturally in optimization and statistics, and also have interest for discrete mathematics because permutation matrices, latin squares, and magic squares appear naturally as lattice points of these polytopes. In this paper we survey advances on the understanding of the combinatorics and geometry of these polyhedra and include some recent unpublished results on the diameter of graphs of these polytopes. In particular, this is a thirty-year update on the status of a list of open questions last visited in the 1984 book by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure

Geometric Combinatorics of Transportation Polytopes and the Behavior of the Simplex Method

This dissertation investigates the geometric combinatorics of convex polytopes and connections to the behavior of the simplex method for linear programming. We focus our attention on transportation polytopes, which are sets of all tables of non-negative real numbers satisfying certain summation conditions. Transportation problems are, in many ways, the simplest kind of linear programs and thus have a rich combinatorial structure. First, we give new results on the diameters of certain classes of transportation polytopes and their relation to the Hirsch Conjecture, which asserts that the diameter of every $d$-dimensional convex polytope with $n$ facets is bounded above by $n-d$. In particular, we prove a new quadratic upper bound on the diameter of $3$-way axial transportation polytopes defined by $1$-marginals. We also show that the Hirsch Conjecture holds for $p \times 2$ classical transportation polytopes, but that there are infinitely-many Hirsch-sharp classical transportation polytopes. Second, we present new results on subpolytopes of transportation polytopes. We investigate, for example, a non-regular triangulation of a subpolytope of the fourth Birkhoff polytope $B_4$. This implies the existence of non-regular triangulations of all Birkhoff polytopes $B_n$ for $n \geq 4$. We also study certain classes of network flow polytopes and prove new linear upper bounds for their diameters.Comment: PhD thesis submitted June 2010 to the University of California, Davis. 183 pages, 49 figure

Graphs of Transportation Polytopes

This paper discusses properties of the graphs of 2-way and 3-way transportation polytopes, in particular, their possible numbers of vertices and their diameters. Our main results include a quadratic bound on the diameter of axial 3-way transportation polytopes and a catalogue of non-degenerate transportation polytopes of small sizes. The catalogue disproves five conjectures about these polyhedra stated in the monograph by Yemelichev et al. (1984). It also allowed us to discover some new results. For example, we prove that the number of vertices of an $m\times n$ transportation polytope is a multiple of the greatest common divisor of $m$ and $n$.Comment: 29 pages, 7 figures. Final version. Improvements to the exposition of several lemmas and the upper bound in Theorem 1.1 is improved by a factor of tw

Solution Approaches to the Three-index Assignment Problem

This thesis explores the axial Three-Index Assignment Problem (3IAP), also called the Multidimensional Assignment Problem. The problem consists in allocating n jobs to n machines in n factories, such that exactly one job is executed by one machine in one factory at a minimum total cost. The 3IAP is an extension of the classical two-dimensional assignment problem. This combinatorial optimisation problem has been the subject of numerous research endeavours, and proven NP-hard due to its inextricable nature. The study adopts an algorithmic approach to develop swift and e ective methods for solving the problem, focusing on balancing computational e ciency and solution accuracy. The Greedy-Style Procedure (GSP) is a novel heuristic algorithm for solving the 3IAP, guaranteeing feasible solutions in polynomial time. Speci c arrangements of cost matrices can lead to the generation of higher-quality feasible solutions. In addressing the 3IAP, analysing the tie-cases and the matrix ordering led to new variants. Further exploration of cost matrix characteristics has allowed two new heuristic classes to be devised for solving 3IAP. The approach focuses on selecting the best solution within each class, resulting in an optimal or a high-quality approximate solution. Numerical experiments con rm the e ciency of these heuristics, consistently delivering quality feasible solutions in competitive computational times. Moreover, by employing diverse optimisation solvers, we propose and implement two e ective methods to achieve optimal solutions for 3IAP in good CPU times. The study introduces two local search methods based on evolutionary algorithms to solve 3IAP. These approaches explore the solution space through random permutations and the Hungarian method. Building on this, a hybrid genetic algorithm that integrates these local search strategies has been proposed for solving the 3IAP. Implementing the Hybrid Genetic Algorithm (HGA) produces high-quality solutions with reduced computational time, surpassing traditional deterministic approaches. The e ciency of the HGA is demonstrated through experimental results and comparative analyses. On medium to large 3IAP instances, our method delivers comparable or better solutions within a competitive computational time frame. Two potential future developments and expected applications are proposed at the end of this project. The rst extension will examine the correlation between cost matrices and the optimal total cost of the assignment and will investigate the dependence structure of matrices and its inuence on optimal solutions. Copula theory and Sklar's theorem can help with this analysis. The focus will be on understanding the stochastic dependence of cost matrices and their multivariate properties. Furthermore, the impact of variations in cost distributions, is often modelled based on economic sectors. The second extension involves integrating variable costs de ned by speci c probability distributions, enhancing the comprehensive analysis of economic scenarios and their impact on the assignment problem. The study considers various well-de ned probability distributions and highlights more practical applications of the assignment problem in real-world economics. The project's original contribution lies in its algorithmic approach to investigating the 3IAP, which has led to the development of new, fast, and e cient heuristic methods that strategically balance computational speed and the accuracy of the solutions achieved

Infinitesimal semi-invariant pictures and co-amalgamation

The purpose of this paper is to study the local structure of the semi-invariant picture of a tame hereditary algebra near the null root. Using a construction that we call co-amalgamation, we show that this local structure is completely described by the semi-invariant pictures of a collection of Nakayama algebras. We then show that this local structure is (piecewise linearly) invariant under cluster tilting.Comment: 40 pages, 6 figure

Integer and Constraint programming methods for mutually Orthogonal Latin Squares.

This thesis examines the Orthogonal Latin Squares (OLS) problem from the viewpoint of Integer and Constraint programming. An Integer Programming (IP) model is proposed and the associated polytope is analysed. We identify several families of strong valid inequalities, namely inequalities arising from cliques, odd holes, antiwebs and wheels of the associated intersection graph. The dimension of the OLS polytope is established and it is proved that certain valid inequalities are facet-inducing. This analysis reveals also a new family of facet-defining inequalities for the polytope associated with the Latin square problem. Separation algorithms of the lowest complexity are presented for particular families of valid inequalities. We illustrate a method for reducing problem's symmetry, which extends previously known results. This allows us to devise an alternative proof for the non-existence of an OLS structure for n = 6, based solely on Linear Programming. Moreover, we present a more general Branch & Cut algorithm for the OLS problem. The algorithm exploits problem structure via integer preprocessing and a specialised branching mechanism. It also incorporates families of strong valid inequalities. Computational analysis is conducted in order to illustrate the significant improvements over simple Branch & Bound. Next, the Constraint Programming (CP) paradigm is examined. Important aspects of designing an efficient CP solver, such as branching strategies and constraint propagation procedures, are evaluated by comprehensive, problem-specific, experiments. The CP algorithms lead to computationally favourable results. In particular, the infeasible case of n = 6, which requires enumerating the entire solution space, is solved in a few seconds. A broader aim of our research is to successfully integrate IP and CP. Hence, we present ideas concerning the unification of IP and CP methods in the form of hybrid algorithms. Two such algorithms are presented and their behaviour is analysed via experimentation. The main finding is that hybrid algorithms are clearly more efficient, as problem size grows, and exhibit a more robust performance than traditional IP and CP algorithms. A hybrid algorithm is also designed for the problem of finding triples of Mutually Orthogonal Latin Squares (MOLS). Given that the OLS problem is a special form of an assignment problem, the last part of the thesis considers multidimensional assignment problems. It introduces a model encompassing all assignment structures, including the case of MOLS. A necessary condition for the existence of an assignment structure is revealed. Relations among assignment problems are also examined, leading to a proposed hierarchy. Further, the polyhedral analysis presented unifies and generalises previous results

Proceedings of the XIII Global Optimization Workshop: GOW'16

[Excerpt] Preface: Past Global Optimization Workshop shave been held in Sopron (1985 and 1990), Szeged (WGO, 1995), Florence (GO’99, 1999), Hanmer Springs (Let’s GO, 2001), Santorini (Frontiers in GO, 2003), San José (Go’05, 2005), Mykonos (AGO’07, 2007), Skukuza (SAGO’08, 2008), Toulouse (TOGO’10, 2010), Natal (NAGO’12, 2012) and Málaga (MAGO’14, 2014) with the aim of stimulating discussion between senior and junior researchers on the topic of Global Optimization. In 2016, the XIII Global Optimization Workshop (GOW’16) takes place in Braga and is organized by three researchers from the University of Minho. Two of them belong to the Systems Engineering and Operational Research Group from the Algoritmi Research Centre and the other to the Statistics, Applied Probability and Operational Research Group from the Centre of Mathematics. The event received more than 50 submissions from 15 countries from Europe, South America and North America. We want to express our gratitude to the invited speaker Panos Pardalos for accepting the invitation and sharing his expertise, helping us to meet the workshop objectives. GOW’16 would not have been possible without the valuable contribution from the authors and the International Scientific Committee members. We thank you all. This proceedings book intends to present an overview of the topics that will be addressed in the workshop with the goal of contributing to interesting and fruitful discussions between the authors and participants. After the event, high quality papers can be submitted to a special issue of the Journal of Global Optimization dedicated to the workshop. [...

The development of a scenario independent method for evaluating the evacuation complexity of a building

Over the past two decades, more than 30 evacuation models have been developed to reproduce people’s movement patterns in evacuation. However, evacuation models cannot assess whether one building is better than another in regards to evacuation wayfinding. There exist techniques that attempt to compare different buildings for evacuation complexity. However, these graph measures are primarily used to measure the relative accessibility of different locations in a spatial system and were not generated for the purpose of comparing the complexity of different buildings. Currently only one method exists, Donegan’s method [DT98] [PD96] [DT99], which can be applied to compare building for evacuation ability. However, this technique is severely limited to specific building layouts and only considers connectivity. Taking the Donegan’s method as a first step, this thesis extends this algorithm to obtain a new Distance Graph Method, which considers travel distance as well as being able to be applied to graphs with circuits. Then a further building complexity measures is presented, the Global Complexity (PAT) method. This is shown to be a valid measure which considers additional important factors such as wayfinding time, travel distance and the areas of compartments. The Distance Graph Method and Global Complexity (PAT) methods are based on a room graph representation which does not have the descriptive power to describe the actual routes taken during the wayfinding process. To resolve this drawback a further method is presented which utilises a ‘route-based graph’ that has the ability to represent the real route that an evacuee will take during the wayfinding process. Furthermore the Distance Graph Method and Global Complexity (PAT) methods assume a “worst state” calculation for the nodal information. This means for buildings with more than one exit these methods calculate a global building complexity according to a mathematical formula, which considers all exits separately. To address these problems, the final method, Complexity Time Measure, is presented, which is based around a number of wayfinding behaviour rules over a ‘route-based graph’ representation. This addresses the question: ‘If an occupant is positioned at a random location within a building, on average how long does the occupant need to spend to find an available exit?’ Hence, provides a means to compare complex buildings, with circuits, in relation to evacuation capability

College of Engineering

Cornell University Courses of Study Vol. 89 1997/9