All-norm approximation algorithms

Yossi Azar, Leah Epstein, Yossi Richter, Gerhard J. Woeginger

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)120-133
Number of pages14
JournalJournal of Algorithms
Issue number2
StatePublished - Aug 2004
Externally publishedYes

Bibliographical note

Funding Information:
* Corresponding author. E-mail addresses: (Y. Azar), (L. Epstein), (Y. Richter), (G.J. Woeginger). 1 Research supported in part by the Israeli Ministry of industry and trade and by the Israel Science Foundation. 2 Research supported in part by the Israel Science Foundation. 3 Research supported in part by the Israeli Ministry of industry and trade. 4 Supported by the START program Y43-MAT of the Austrian Ministry of Science.

ASJC Scopus subject areas

  • Control and Optimization
  • Computational Mathematics
  • Computational Theory and Mathematics


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  • All-norm approximation algorithms

    Azar, Y., Epstein, L., Richter, Y. & Woeginger, G. J., 2002, Algorithm Theory - SWAT 2002 - 8th Scandinavian Workshop on Algorithm Theory, Proceedings. Penttonen, M. & Schmidt, E. M. (eds.). Springer Verlag, p. 288-297 10 p. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); vol. 2368).

    Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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