The fractional chromatic number of triangle-free subcubic graphs

Heckman and Thomas conjectured that the fractional chromatic number of any triangle-free subcubic graph is at most 14/5. Improving on estimates of Hatami and Zhu and of Lu and Peng, we prove that the fractional chromatic number of any triangle-free subcubic graph is at most 32/11 (which is roughly 2.909)

Topics in graph colouring and graph structures

This thesis investigates problems in a number of different areas of graph theory. These problems are related in the sense that they mostly concern the colouring or structure of the underlying graph. The first problem we consider is in Ramsey Theory, a branch of graph theory stemming from the eponymous theorem which, in its simplest form, states that any sufficiently large graph will contain a clique or anti-clique of a specified size. The problem of finding the minimum size of underlying graph which will guarantee such a clique or anti-clique is an interesting problem in its own right, which has received much interest over the last eighty years but which is notoriously intractable. We consider a generalisation of this problem. Rather than edges being present or not present in the underlying graph, each is assigned one of three possible colours and, rather than considering cliques, we consider cycles. Combining regularity and stability methods, we prove an exact result for a triple of long cycles. We then move on to consider removal lemmas. The classic Removal Lemma states that, for n sufficiently large, any graph on n vertices containing o(n^3) triangles can be made triangle-free by the removal of o(n^2) edges. Utilising a coloured hypergraph generalisation of this result, we prove removal lemmas for two classes of multinomials. Next, we consider a problem in fractional colouring. Since finding the chromatic number of a given graph can be viewed as an integer programming problem, it is natural to consider the solution to the corresponding linear programming problem. The solution to this LP-relaxation is called the fractional chromatic number. By a probabilistic method, we improve on the best previously known bound for the fractional chromatic number of a triangle-free graph with maximum degree at most three. Finally, we prove a weak version of Vizing's Theorem for hypergraphs. We prove that, if H is an intersecting 3-uniform hypergraph with maximum degree D and maximum multiplicity m, then H has at most 2D+m edges. Furthermore, we prove that the unique structure achieving this maximum is m copies of the Fano Plane

Facility location and related problems

PRINTAUSGABE IN HAUPTBIBLIOTHEK NICHT EINGELANGT! -- Bei Standortoptimierungsproblemen geht es um eine strategisch günstige Auswahl von Orten unter den Gesichtspunkten des Nutzens und der Aufwände, die mit den Standort-entscheidungen einhergehen. Beispielsweise können in der Planung die lageabhängigen Betriebskosten und die Errichtungskosten gegeneinander aufgewogen werden. Der zentrale Beitrag der vorliegenden Arbeit sind zwei Erweiterungen von Standortproblemen die durch einen Überblick klassischer Modelle eingefasst werden. Die eine Erweiterung behandelt ein dynamisches Warehouse-Location Problem in einem stochastischen Umfeld: Während mehrerer Perioden können Standorte geöffnet und geschlossen werden. Ziel ist die Minimierung der erwarteten Kosten die sich aus Betriebskosten, Produktionskosten, Transportkosten, Lagerhaltungskosten und Strafkosten bei Fehlmengen zusammensetzen. Ein exaktes und ein heuristisches Lösungsverfahren werden vorgestellt. Die zweite Erweiterung kann man als doppeltes Set-Cover Problem verstehen. Es sollen Kunden mit zwei Dienstleistungen bedient werden, die an Zentren gebunden sind. Jeder Kunde muss von mindestens einem Zentrum eines jeden Dienstleistungstyps erreichbar sein. Gleichzeitig ist darauf zu achten, dass die Anzahl verwendeter Zentren beschränkt ist und dass die Zentren höchstens einer Dienstleistung zugeordnet sind. Es werden verschiedene Anwendungen vorgestellt, und durch Einschränkungen wird versucht die Grenze zwischen Problemen mit polynomiellem Aufwand und NP-schweren Problemen zu ziehen. Im Rahmen einer bioinformatischen Anwendung wird eine Ant-Colony Metaheuristik eingesetzt.Facility location treats the problem of choosing locations while respecting effort and utility. E.g.: we can think of balancing the maintenance and setup costs for a facility. The central contribution of this work are two extensions of classical location models that get enclosed into the presentation of standard facility location models. One of the extensions is a dynamic warehouse location problem in a stochastic environment. Within a planning horizon of given number of periods we are able to open and close facilities and the aim is to minimize the expected costs. The costs consist of operating costs, production costs, inventory costs and penalty costs for shortages. We present an exact method and a heuristic approach. The second extension can be regarded as a double Set Cover Problem. We have to maintain two services by allocating corresponding sites and each customer has to be reachable by at least one of the centers and each service type. Simultaneously we have to respect that the number of used locations is limited, while no location is assigned to two services. We present different applications and by restricting the problem we draw the line between polynomially solvable problems and intractable ones. In the context of an application in bio-informatics we develop an ACO heuristic

Large-scale mixed integer optimization approaches for scheduling airline operations under irregularity

Perhaps no single industry has benefited more from advancements in computation, analytics, and optimization than the airline industry. Operations Research (OR) is now ubiquitous in the way airlines develop their schedules, price their itineraries, manage their fleet, route their aircraft, and schedule their crew. These problems, among others, are well-known to industry practitioners and academics alike and arise within the context of the planning environment which takes place well in advance of the date of departure. One salient feature of the planning environment is that decisions are made in a frictionless environment that do not consider perturbations to an existing schedule. Airline operations are rife with disruptions caused by factors such as convective weather, aircraft failure, air traffic control restrictions, network effects, among other irregularities. Substantially less work in the OR community has been examined within the context of the real-time operational environment. While problems in the planning and operational environments are similar from a mathematical perspective, the complexity of the operational environment is exacerbated by two factors. First, decisions need to be made in as close to real-time as possible. Unlike the planning phase, decision-makers do not have hours of time to return a decision. Secondly, there are a host of operational considerations in which complex rules mandated by regulatory agencies like the Federal Administration Association (FAA), airline requirements, or union rules. Such restrictions often make finding even a feasible set of re-scheduling decisions an arduous task, let alone the global optimum. The goals and objectives of this thesis are found in Chapter 1. Chapter 2 provides an overview airline operations and the current practices of disruption management employed at most airlines. Both the causes and the costs associated with irregular operations are surveyed. The role of airline Operations Control Center (OCC) is discussed in which serves as the real-time decision making environment that is important to understand for the body of this work. Chapter 3 introduces an optimization-based approach to solve the Airline Integrated Recovery (AIR) problem that simultaneously solves re-scheduling decisions for the operating schedule, aircraft routings, crew assignments, and passenger itineraries. The methodology is validated by using real-world industrial data from a U.S. hub-and-spoke regional carrier and we show how the incumbent approach can dominate the incumbent sequential approach in way that is amenable to the operational constraints imposed by a decision-making environment. Computational effort is central to the efficacy of any algorithm present in a real-time decision making environment such as an OCC. The latter two chapters illustrate various methods that are shown to expedite more traditional large-scale optimization methods that are applicable a wide family of optimization problems, including the AIR problem. Chapter 4 shows how delayed constraint generation and column generation may be used simultaneously through use of alternate polyhedra that verify whether or not a given cut that has been generated from a subset of variables remains globally valid. While Benders' decomposition is a well-known algorithm to solve problems exhibiting a block structure, one possible drawback is slow convergence. Expediting Benders' decomposition has been explored in the literature through model reformulation, improving bounds, and cut selection strategies, but little has been studied how to strengthen a standard cut. Chapter 5 examines four methods for the convergence may be accelerated through an affine transformation into the interior of the feasible set, generating a split cut induced by a standard Benders' inequality, sequential lifting, and superadditive lifting over a relaxation of a multi-row system. It is shown that the first two methods yield the most promising results within the context of an AIR model.PhDCommittee Co-Chair: Clarke, John-Paul; Committee Co-Chair: Johnson, Ellis; Committee Member: Ahmed, Shabbir; Committee Member: Clarke, Michael; Committee Member: Nemhauser, Georg

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum