On Combinatorial Search Problems Which Involve Graphs

Preface Reality provides many examples of something we call a search. Informally, we may have a goal (e.g., finding a lost key, a damaged light bulb, the shortest way to the next bus stop) and a set of possible actions to gain some information. All possible action needs some time, money, work etc., so that we want to calculate the expenses of our search strategy in order to minimize them. Due to their ubiquity and their variety, search problems are subject to an enormous amount of modeling methods. A general and very simple method arises if the " set of actions " can be made discrete, that is, we have a finite number of things we can do (e.g., to ask someone for the way, to see if the key is inside some object like a pocket, change a bulb), maybe separated from a finite number of objects to which we can apply our actions (persons, boxes, bulbs). Furthermore, we have a finite number of possible results to each action, at least two (knows – does not know, is inside – is not, works or doesn't). In this way, we collect information until we have reached the goal. Often, we can simplify the computation of the expenses just counting the number of steps we need. This is the starting point of combinatorial search. We call the set of " things " we can do the set of questions and the results answers. Since the outcome of our search is unknown, we may want to find a bound to the expenses of the worst case or to calculate an average value if we know enough about the probabilities of the respective results. With our model we ensure that there are only finitely many questions necessary to finish the search. So the expenses of the worst case are finite. The set or tupel of answers we obtain during the search provides us with some information. Our goal, finally, has to be represented by a finite set of search results (all pockets, all sockets, every sequence of streets of a city) from which one element is selected (e.g., the pocket with the key, the socket with the damaged light bulb, the shortest way). For information theory says: there has to be a unique tupel of answers to each of the possible search results. Hence, a lower bound of the required number of questions …