Chapter 1 MULTI INDEX ASSIGNMENT PROBLEMS : COMPLEXITY , APPROXIMATION , APPLICATIONS

This chapter deals with approximation algorithms for and applications of multi index assignment problems (MIAPs). MIAPs and relatives of it have a relatively long history both in applications as well as in theoretical results, starting at least in the fifties (see e.g. Motzkin, 1952, Schell, 1955 and Koopmans and Beckmann, 1957). Here we intend to give the reader i) an idea of the range and diversity of practical problems that have been formulated as an MIAP, and ii) an overview on what is known on theoretical aspects of solving instances of MIAPs. In particular, we will discuss complexity and approximability issues for special cases of MIAPs. We feel that investigating special cases of MIAPs is an important topic since real-world instances almost always posses a certain structure that can be exploited when it comes to solving them. Before doing so, let us first describe a somewhat frivolous, quite unrealistic situation that gives rise to an instance of an MIAP. Suppose that you have become mayor of a set of villages. Everything is running smoothly, except in one village where chaos seems to rule. You, as a responsible mayor should, start looking into this and after careful gathering of information you find that there 30 cats, 30 houses, 30 men and 30 women present in the village. Now, in order to establish some peace you impose that 30 "units" will have to be formed, each consisting of a cat, a house, a man, and a woman. It turns out that for each possible 4-tuple (and there are 304 =810000 of them), a number is known that reflects the happiness of this particular unit. Since you as a mayor