We consider a random assignment problem in an m by n matrix A , where some of the entries are zero, and the others are independent exponentially distributed variables with arbitrary mean values. We want to determine the expected value of the optimal k -assignment. I will report on some recent progress on the problem. We have been able to answer the more general problem in the special case when the set of zeros contains a k -1 assignment. In two relatively recent papers, M. M. Paulhus and A. Chalcraft consider the problem of packing an infinite collection of square tiles of finite total area into a rectangular box of the same area as the total area of the squares.
Paulhus paper describes a strategy for packing the squares which is simple and quite efficient.
Chalcraft introduces methods for proving that perfect packings exist in some specific cases. He shows that for certain exponents in the interval , it is possible to pack this set of squares into a square box of the same area as the total area of the squares, and remarks that an interesting challenge is to try to obtain such results for larger values of t . Each lamp toggles a subset of the lamps. The player aims at lighting every lamp. Games of this type have been around since the sixties.