Approximating robust bin-packing with budgeted uncertainty

Marin Bougeret (1) , Noam Goldberg (2) , Michael Poss (3)

1 ALGCO - Algorithmes, Graphes et Combinatoire
2 Bar-Ilan University [Israël]
3 MAORE - Methods, Algorithms for Operations REsearch

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 ⊆ × n i=1 [ai, ai + ˆ ai], where a ∈ [0, 1] n represents the nominal sizes of the items andâandˆandâ ∈ [0, 1] n their possible deviations. We consider more specifically two uncertainty sets previously studied in the literature. The first set, denoted U Γ , contains scenarios in which at most Γ ∈ N items deviate, each of them reaching its peak value ai + ˆ ai, while each other item has its nominal value ai. The second set, denoted U Ω , bounds by Ω ∈ [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 Ω , and a 2(Γ +1) approximation for model U Γ (which can be improved to 2 approximation for Γ = 1). This motivates the question of the existence of a constant ratio approximation algorithm for the U Γ model. Our main result is to answer positively to this question by providing a 4.5 approximation for U Γ model based on dynamic programming.

Domaines: Recherche opérationnelle [math.OC]

Marin Bougeret, Noam Goldberg, Michael Poss. Approximating robust bin-packing with budgeted uncertainty. 2020. ⟨hal-02119351v2⟩