Complexity to Find Wiener Index of Some Graphs

The Wiener index is one of the oldest graph parameter which is used to study molecular-graph-based structure. This parameter was first proposed by Harold Wiener in 1947 to determining the boiling point of paraffin. The Wiener index of a molecular graph measures the compactness of the underlying molecule. This parameter is wide studied area for molecular chemistry. It is used to study the physio-chemical properties of the underlying organic compounds. The Wiener index of a connected graph is denoted by W(G) and is defined as, that is W(G) is the sum of distances between all pairs (ordered) of vertices of G. In this paper, we give the algorithmic idea to find the Wiener index of some graphs, like cactus graphs and intersection graphs, viz. interval, circular-arc, permutation, trapezoid graphs.Comment: 6 page

Scheduling algorithm to select kk optimal programme slots in television channels: A graph theoretic approach

In this paper, it is shown that all programmes of all television channels can be modelled as an interval graph. The programme slots are taken as the vertices of the graph and if the time duration of two {programme slots} have non-empty intersection, the corresponding vertices are considered to be connected by an edge. The number of viewers of a programme is taken as the weight of the vertex. A set of programmes that are mutually exclusive in respect of time scheduling is called a session. We assume that a company sets the objective of selecting the popular programmes in $k$ parallel sessions among different channels so as to make its commercial advertisement reach the maximum number of viewers, that is, a company selects $k$ suitable programme slots simultaneously for advertisement. The aim of the paper is, therefore, to {help} the companies to select the programme slots, which are mutually exclusive with respect to the time schedule of telecasting time, in such a way that the total number of viewers of the selected programme in $k$ parallel slots rises to the optimum level. It is shown that the solution of this problem is obtained by solving the maximum weight $k$-colouring problem on an interval {graph}. An algorithm is designed to solve this just-in-time optimization problem using $O(kMn^2)$ time, where $n$ and $M$ represent the total number of programmes of all channels and the upper bound of the viewers of all programmes of all channels respectively. The problem considered in this paper is a daily life problem which is modeled by $k$-colouring problem on interval graph.Comment: 25 page

L(2,1)-labelling of Circular-arc Graph

An L(2,1)-labelling of a graph $G=(V, E)$ is $\lambda_{2,1}(G)$ a function $f$ from the vertex set V (G) to the set of non-negative integers such that adjacent vertices get numbers at least two apart, and vertices at distance two get distinct numbers. The L(2,1)-labelling number denoted by $\lambda_{2,1}(G)$ of $G$ is the minimum range of labels over all such labelling. In this article, it is shown that, for a circular-arc graph $G$, the upper bound of $\lambda_{2,1}(G)$ is $\Delta+3\omega$, where $\Delta$ and $\omega$ represents the maximum degree of the vertices and size of maximum clique respectively.Comment: 12 page

Revenue management models for hotel business

Competition and macroeconomic changes stimulate hoteliers to find ways to improve business. Hotel Revenue Management (HRM) techniques evolve and their beneficial results attract more and more hotel owners. Solutions of hotel revenue management approaches support managers in decision-making and increase revenues. In order to contribute to HRM theory and practice the following two problems of revenue management in hotel business have been studied: 1) problem P-Pricing, which is a dynamic and uncertain problem of determining prices of rooms of different categories such that the total profit of room sales based on the forecasted demand is maximized, assuming that the demand is price sensitive, and 2) problem P-Select, which is a static and deterministic problem of selecting a subset of room requests from a given set of room requests such that the selected requests can be assigned to physically different rooms or the same room in different time slots and the total value of these requests is maximized. In problem P-Pricing specific historical forecasting methods are used to predict values of coefficients of linear demand function. Revenue maximization of a hotel is gained via solving mathematical programming problem with concave quadratic objective function and linear constraints. A typical example of a practical situation where problem P-Pricing appears is the reservation of hotel rooms via an Internet service, which immediately accepts a request if it can be satisfied. Problem P-Select is modeled as a Fixed Interval Scheduling Problem on parallel machines. Its relation to the Maximum Weight Clique Problem of graph theory is established. Optimal and heuristic solution approaches are developed and computer tested. A typical example of a practical situation where problem P-Select appears is renting of private apartments and cottages, when the owner collects requests during a certain period of time and then decides which of them to accept. It also appears in hotel business in special cases such as world sporting events, when requests for rooms come much in advance, demand is usually exaggerated and hotel managers have the possibility to consider requests during a certain period of time.Der zunehmende Wettbewerb im Hotelgewerbe zwingt Hoteliers zum Einsatz von Entscheidungsunterstützungssoftware, die eine flexiblere, nachfrageorientiertere Preispolitik auf Basis der bei Flugbuchungen bekannten Methoden des Revenue Managements erfordert. Diese Dissertation betrachtet die beiden Probleme der Bestimmung des geeigneten Zimmerpreises jeder Zimmerkategorie, bei zu großer, auf Vergangenheitsdaten basierter, prognostizierter Nachfrage (P-Pricing), sowie das Problem der Auswahl von Nachfragen (Hotelbuchungen) bei vorliegender Überbuchung (P-Select). P-Pricing ist ein dynamisches, auf unsicheren Daten basierendes Gewinnmaximierungsproblem bei preiselastischer Nachfrage. P-Select ist ein deterministisches Problem zur gewinnmaximalen Auswahl von Zimmernachfragen, die sich auf entweder unterschiedliche Zeiträume für gleiche Zimmer oder gleiche Zeiträume für unterschiedliche Zimmer beziehen. P-Pricing wird als mathematisches Programmierungsproblem mit konkaver, quadratischer Zielfunktion und linearen Restriktionen modelliert. Die zu erwartende Nachfrage wird über die vorliegenden Vergangenheitswerte prognostiziert. Beispielhaft ist hier etwa die Buchen eines Hotelzimmers über ein Internetportal, das eine Nachfrage sofort akzeptiert, sofern sie befriedigt werden kann. Eine Preisanpassung findet dann unmittelbar statt. P-Select wird als Intervall Scheduling Problem auf parallelen Maschinen mit festen Intervallen formuliert und läuft darauf hinaus, in einem Graphen maximale, gewichtete Cliquen zu bestimmen. Exakte und heuristische Verfahren werden vorgestellt und getestet. Beispielhaft für Problem P-Select steht die Vermietung von privaten Wohnungen. Der Besitzer sammelt für einen gewissen Zeitraum alle eingehenden Anfrage und wählt anschließend die für Ihn geeignetsten aus. Die Situation tritt im Hotelgewerbe eher dann auf, wenn aufgrund von Großveranstaltungen auf knappe Hotelressourcen zurückgegriffen wird, die bereits frühzeitig angefragt werden. Der Arbeit liegen reale Daten zugrunde