We study the following problem, introduced by in 2006. We are given, online or offline, a set of coloured items of different sizes, and wish to pack them into bins of equal size so that we use few bins in total (at most α times optimal), and that the items of each colour span few bins (at most β times optimal). We call such allocations ( α , β ) -approximate. As usual in bin packing problems, we allow additive constants and consider ( α , β ) as the asymptotic performance ratios. We prove that for e 0 , if we desire small α, no scheme can beat ( 1 + e , ? ( 1 e ) ) -approximate allocations and similarly as we desire small β, no scheme can beat ( 1.69103 , 1 + e ) -approximate allocations. We give offline schemes that come very close to achieving these lower bounds. For the online case, we prove that no scheme can even achieve ( O ( 1 ) , O ( 1 ) ) -approximate allocations. However, a small restriction on item sizes permits a simple online scheme that computes ( 2 + e , 1.7 ) -approximate allocations.
@cite_15 consider the case where each item is of a different colour and can be fractionally (arbitrarily) divided between bins, bins have different sizes and the total weight of items exactly equals the total weight of bins. They show how to compute an allocation that is asymptotically optimal for each colour. By contrast, we relax the assumption that we must exactly fill all the bins, and consider the case of indivisible allocations. In this setting, the problem is much more interesting: it is impossible to get arbitrarily good @math -approximate allocations in general. Thus, these relaxed packings have a tradeoff between bin stretch and colour stretch, with polynomial-time approximations. We also consider for the first time the case where items arrive online. However, the case of heterogenous bins is open for our setting.
@cite_15
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In this paper, we consider the variable sized bin packing problem where the objective is to minimize the total cost of used bins when the cost of unit size of each bin does not increase as the bin size increases. Two greedy algorithms are described, and analyzed in three special cases: (a) the sizes of items and bins are divisible, respectively, (b) only the sizes of bins are divisible, and (c) the sizes of bins are not divisible. Here, we say that a list of numbers a1,a2,,am are divisible when aj exactly divides aj1, for each 1<jm. In the case of (a), the algorithms give optimal solutions, and in the case of (b), each algorithm gives a solution whose value is less than 119C(B*)+4119, where C(B*) is the optimal value. In the case of (c), each algorithm gives a solution whose value is less than 32C(B*)+1.