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 two uncertainty sets previously studied in literature. The first is budgeted (the $U^\Gamma$ model), which at most $\Gamma$ items deviate, each reaching its peak value, while other assume their nominal values. second set, $U^\Omega$ model, bounds total amount deviation scenario. show that a variant Next-cover algorithm $2$ approximation for and another this $2\Gamma$ model. Unlike classical problem, it shown (unless $\mathcal{P}=\mathcal{NP}$) no asymptotic scheme exists $\Gamma=1$. This motivates question existence constant factor Our main result to answer by proving (polynomial-time) $4.5$ algorithm, based on dynamic-programming approach.