The Bin Packing problem takes as input an infinite supply of bins B_1, B_2, B_3, ..., each bin of size 1 unit. We are also given n items a_1, a_2, ...a_n, ..., and and each item has a size S_j that is a real number in the interval [0, 1], The Bin Packing problem seeks to find the smallest number of bins such that all n items can be packed into these bins. For example, suppose that we are given 4 items a_1, a_2, a_3 and a_4 of sizes 0.5, 0.4, 0.6, and 0.5 respectively. We could pack a_1 and a_2 in bin Bx because s_1 + s_2 = 0.9 lessthanorequalto 1. We could then pack a_3 into bin B_2, but we could not also add a_4 to bin B_2, because S_3 + S_4 = 1.1 > 1. So a4 would have to be packed in bin B_3. This gives us a bin packing of the 4 items into three bins. An alternate way of packing items that would lead to the use of just two bins is to pack a_1 and a_4 into bin B_1 and