Approximation results for makespan minimization with budgeted uncertainty

International audienceWe study approximation algorithms for the problem of minimizing the makespan on a set of machines with uncertainty on the processing times of jobs. In the model we consider, which goes back to [3], once the schedule is defined an adversary can pick a scenario where deviation is added to some of the jobs' processing times. Given only the maximal cardinality of these jobs, and the magnitude of potential deviation for each job, the goal is to optimize the worst-case scenario. We consider both the cases of identical and unrelated machines. Our main result is an EPTAS for the case of identical machines. We also provide a 3-approximation algorithm and an inapproximability ratio of 2 − epsilon for the case of unrelated machines

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 ⊆ × 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