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 $\alpha$ times optimal), and that the items of each colour span few bins (at most $\beta$ times optimal). We call such allocations $(\alpha, \beta)$-approximate. As usual in bin packing problems, we allow additive constants and consider $(\alpha,\beta)$ as the asymptotic performance ratios. We prove that for $\eps>0$, if we desire small $\alpha$, no scheme can beat $(1+\eps, \Omega(1/\eps))$-approximate allocations and similarly as we desire small $\beta$, 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.