2. Classic Beginnings

The central theme of this book is the ability to identify the best location of one or more facilities or objects in order to provide some type or level of coverage. For example, let us suppose that we wish to place guards in an art gallery in such a manner that all areas of the gallery are within sight of one or more guards. In essence, we want the set of guard positions to “cover” or view the entire public area of the gallery. There can, of course, be many different configurations in which guards can view the entire gallery; however, it is of practical necessity to seek a pattern that deploys the fewest number of guards. As a second example, consider the case where we desire to provide fire protection to all neighborhoods of a city. In order to respond to a fire in a timely manner, we may set a standard that each neighborhood of the city should be no more than a mile and a half away from their nearest fire station (where fire trucks and crews can be housed to quickly respond when called). The fire service deployment problem can then be defined as finding the fewest fire stations (and their locations) so that each neighborhood is served or covered within a mile and a half of a station. Both the gallery guard positioning problem and the fire station location problem are examples of the Location Set Covering Problem, one of many covering problems that will be addressed in this book.