Does Randomization Help in On-Line Bin Packing?

In bin packing problems, a list of items is to be packed into a minimum number of bins of uniform size. We consider the class of on-line algorithms, in which we are given the items sequentially and we must pack the current item before we see subsequent items. There has been much recent interest in various on-line bin packing problems. In [l-4,6] lower bounds have been proved for deterministic algorithms for different bin packing problems. In this note we show that the same lower bounds hold for randomized algorithms. We present some definitions which are common to all the bin packing problems we consider. In later sections we will present more specific definitions. A list L is a sequence of items, aI, a2,. . . , a,,, which have to be packed into uniform size bins. Let A be a deterministic on-line algorithm and let N,(L) be the number of bins used by A when packing L in the given order, i.e., first a, is put in a bin, then a2 is put in a bin, etc., and finally a,, is put in a bin. Let L* be the number of bins used in the optima1 packing of L. The packing ratio r,(L) is N,(L)/L*. Let PC be the set of instances L for which L* 2 c (for