Windows scheduling as a restricted version of bin packing

Given is a sequence of n positive integers w 1 , w 2 ,…, w n that are associated with the items 1,2,… n , respectively. In the windows scheduling problem, the goal is to schedule all the items (equal-length information pages) on broadcasting channels such that the gap between two consecutive appearances of page i on any of the channels is at most w i slots (a slot is the transmission time of one page). In the unit-fractions bin packing problem, the goal is to pack all the items in bins of unit size where the size (width) of item i is 1/ w i . The optimization objective is to minimize the number of channels or bins. In the offline setting, the sequence is known in advance, whereas in the online setting, the items arrive in order and assignment decisions are irrevocable. Since a page requires at least 1/ w i of a channel's bandwidth, it follows that windows scheduling without migration (i.e., all broadcasts of a page must be from the same channel) is a restricted version of unit-fractions bin packing. Let H = ⌈Σ i ==1 n (1/ w i ) be the bandwidth lower bound on the required number of bins (channels). The best-known offline algorithm for the windows scheduling problem used H + O (ln H ) channels. This article presents an offline algorithm for the unit-fractions bin packing problem with at most H + 1 bins. In the online setting, this article presents algorithms for both problems with H + O (√ H ) channels or bins, where the one for the unit-fractions bin packing problem is simpler. On the other hand, this article shows that already for the unit-fractions bin packing problem, any online algorithm must use at least H +Ω(ln H ) bins. For instances in which the window sizes form a divisible sequence, an optimal online algorithm is presented. Finally, this article includes a new NP-hardness proof for the windows scheduling problem.