Approximating Robust Bin Packing with Budgeted Uncertainty

Abstract

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

This research has benefited from the support of the ANR project ROBUST [ANR-16-CE40-0018].