Selection of programme slots of television channels for giving advertisement: A graph theoretic approach

The present paper aims at developing a linear time algorithm to find a solution to the ‘maximum weight 1 colouring’ problem for an interval graph with interval weight. This algorithm has been applied to solve the problem that involves selecting different programme slots telecast on different television channels in a day so as to reach the maximum number of viewers. It is shown that all programmes of all television channels can be modelled as a weighted interval graph with interval numbers as weights. The programme slots are taken as the vertices of the graph and if the time durations 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. In reality, the number of viewers of a programme is not a fixed one; generally, it lies in an interval. Thus, the weights of the vertices are taken as interval numbers. We assume that a company sets the objective of selecting the popular programme in different channels so as to make its commercial advertisement reach the maximum number of viewers. However, the constraint imposed is that all selected programmes are mutually exclusive in respect of time scheduling. The objective of the paper is, therefore, to helps 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 slots rises to the optimum level. It is shown that the solution of this problem is obtained by solving the maximum weight colouring problem on an interval graph. An algorithm is designed to solve this optimization problem using O(n) time, where n represents the total number of programmes of all channels. 2007 Elsevier Inc. All rights reserved. MSC: Primary codes: 05C15; 05C17; 05C85; 05C90; Secondary codes: 68R10; 68W05