Fast approximate PCPs for multidimensional bin-packing problem

We consider approximate PCPs for multidimensional bin-packing problems. In particular, we show how a verifier can be quickly convinced that a set of multidimensional blocks can be packed into a small number of bins. The running time of the verifier is bounded by O(T(n)), where T(n) is the time required to test for heaviness. We give heaviness testers that can test heaviness of an element in the domain [1, ...,n]d in time O((log n)d). We also also give approximate PCPs with efficient verifiers for recursive bin packing and multidimensional routing.