Locality-preserving allocations Problems and coloured Bin Packing

We study the following problem, introduced by Chung et al. 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 $\eps>0$, if we desire small $α$, no scheme can beat $(1+\eps, Ω(1/\eps))$-approximate allocations and similarly as we desire small $β$, no scheme can beat $(1.69103, 1+\eps)$-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 $(2+\eps, 1.7)$-approximate allocations.