All-norm approximation algorithms


Azar, Yossi and Epstein, Leah and Richter, Yossi and Woeginger, Gerhard J. (2004) All-norm approximation algorithms. Journal of Algorithms, 52 (2). pp. 120-133. ISSN 0196-6774

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Abstract: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 ℓp 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 ℓp norms simultaneously. Specifically, we consider the problem of scheduling in the restricted assignment model, where there are m machines and n jobs, each job 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. [Math. Program. 46 (1990) 259–271] showed a 2-approximation algorithm for the problem with respect to the ℓ∞ norm. For any fixed ℓp 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 ℓp norm (p>1) there is no PTAS unless P=NP by showing an APX-hardness result. We also show for any given ℓp norm a FPTAS for any fixed number of machines.
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Copyright:© 2004 Elsevier
Electrical Engineering, Mathematics and Computer Science (EEMCS)
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