Constant-Ratio Approximation for Robust Bin Packing with Budgeted Uncertainty

We consider robust variants of the bin packing problem with uncertain item sizes. Specifically we consider two uncertainty sets previously studied in the literature: budgeted uncertainty (the $\UG$ model) in which at most $\Gamma$ items deviate, each reaching its peak value, while other items assume their nominal values. The second uncertainty set, the $\UO$ model, bounds the total amount of deviation in each scenario. We show that a variant of the next-fit-decreasing algorithm is a $2$ approximation for the $\UO$ model, and another variant of this algorithm is a $2\Gamma$ approximation for the $\UG$ model. Unlike the classical bin packing problem, it is shown that (unless $\PP=\NP$) no asymptotic approximation scheme exists for the $\UG$ model, already for $\Gamma=1$. This motivates the question of the existence of a constant approximation factor algorithm for the $\UG$ model. Our main result is to answer this question by proving a (polynomial-time) $4.5$ approximation algorithm, based on a dynamic-programming approach.