Approximating Robust Bin Packing with Budgeted Uncertainty.

We consider robust variants of the bin-packing problem where the sizes of the items can take any value in a given uncertainty set U subset of xn(i=1)[(a) over bar (i), (a) over bar (i) + (a) over cap (i)], where (a) over bar is an element of [0, 1](n) represents the nominal sizes of the items and (a) over cap is an element of [0, 1](n) their possible deviations. We consider more specifically two uncertainty sets previously studied in the literature. The first set, denoted U-Gamma, contains scenarios in which at most Gamma is an element of N items deviate, each of them reaching its peak value (a) over bar (i) + (a) over cap (i), while each other item has its nominal value (a) over bar (i). The second set, denoted U-Omega, bounds by Omega is an element of [0, 1] the total amount of deviation in each scenario. We show that a variant of the next-fit algorithm provides a 2-approximation for model U-Omega, and a 2(Gamma + 1) approximation for model U-Gamma (which can be improved to 2 approximation for Gamma = 1). This motivates the question of the existence of a constant ratio approximation algorithm for the U-Gamma model. Our main result is to answer positively to this question by providing a 4.5 approximation for U-Gamma model based on dynamic programming.