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 associatedwith 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] showeda 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 restrictedassignment 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 fixedn umber of machines.