 
Summary: Allnorm Approximation Algorithms
Yossi Azar # Leah Epstein + Yossi Richter #
Gerhard J. Woeginger §
Abstract
A major drawback in optimization problems and in particular in scheduling prob
lems is that for every measure there may be a di#erent optimal solution. In many cases
the various measures are di#erent # p norms. We address this problem by introducing
the concept of an Allnorm #approximation algorithm, which supplies one solution that
guarantees #approximation to all # 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 sep
arately. Lenstra et al. [11] showed a 2approximation algorithm for the problem with
respect to the # # norm. For any fixed # p norm the previously known approximation
algorithm has a performance of #(p). We provide an allnorm 2approximation poly
nomial algorithm for the restricted assignment problem. On the other hand, we show
that for any given # p norm (p > 1) there is no PTAS unless P=NP by showing an APX
hardness result. We also show for any given # p norm a FPTAS for any fixed number of
machines.
1 Introduction
