Colored Bin Packing

In this version of Colored Bin-Packing, we have n items of unit weight, where each item is one of x colors, where x ≥ 3. Again, we have an unlimited supply of bins, each with weight limit L, in which to pack the items and our goal is to minimize the total number of bins. 2 Algorithm The unit-weight case introduces weight constraints. Furthermore, we know that color constraints, specifically the discrepancy, forces us to use more bins than the weight constraint alone. The first half of the algorithm depends on discrepancy. If discrepancy is not a problem (i.e. is not more than 0), we simply order and split the bins up similar to the zero-weight case. If discrepancy is an issue (i.e. is more than 0), we handle the bins differently depending on if bin capacity is even or odd. In both cases we pack bins by alternating between items of the most frequent color and items of other colors. If bin capacity is even, we may be able to condense bins by using items of other colors that top some bins to merge bins containing only a single item of the most frequent color. If bin capacity is odd, we check if we can eventually reduce discrepancy to zero. This is because we use one additional max-color item for every bin we pack. If discrepancy eventually becomes zero, we are in the case where discrepancy is not a problem, and apply that algorithm. If this is not so, we will end up with bins topped by items of the most frequent color. The final set will be optimal, since all the bins will be topped with items of the this color and therefore cannot be combined. The Unit-Weight algorithm solves this problem. We first note that we must use at least n L number of bins. Let D denote the discrepancy, or the difference between the number of MaxColor items and OtherColors items. By the previous proof, we know that for 0-weight items, we will either need one bin (in the case where D ≤ 0), or D bins (in the case where D > 0).