All-Norm Approximation Algorithms

A major drawback in optimization problems and in particular in scheduling problems is that for every measure there may be a different optimal solution. In many cases the various measures are different ℓ p norms. We address this problem by introducing the concept of an All-norm ρ-approximation algorithm, which supplies one solution that guarantees ρ-approximation to all ℓ p norms simultaneously. Specifically, we consider the problem of scheduling in the restricted assignment model, where there are m machines and n jobs, each is associated with a subset of the machines and should be assigned to one of them. Previous work considered approximation algorithms for each norm separately. Lenstra et al. [12] showed a 2-approximation algorithm for the problem with respect to the ℓ∞ norm. For any fixed ℓ p norm the previously known approximation algorithm has a performance of θ(p). We provide an all-norm 2-approximation polynomial algorithm for the restricted assignment problem. On the other hand, we show that for any given l p norm (p > 1) there is no PTAS unless P=NP by showing an APX-hardness result. We also show for any given ℓ p norm a FPTAS for any fixed number of machines.