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)(α,β)-approximate. As usual in bin packing problems, we allow additive constants and consider (α,β)(α,β) as the asymptotic performance ratios. We prove that for ε>0ε>0, if we desire small α  , no scheme can beat (1+ε,Ω(1/ε))(1+ε,Ω(1/ε))-approximate allocations and similarly as we desire small β  , no scheme can beat (1.69103,1+ε)(1.69103,1+ε)-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))(O(1),O(1))-approximate allocations. However, a small restriction on item sizes permits a simple online scheme that computes (2+ε,1.7)(2+ε,1.7)-approximate allocations.