TY - JOUR
T1 - All-norm approximation algorithms
AU - Azar, Yossi
AU - Epstein, Leah
AU - Richter, Yossi
AU - Woeginger, Gerhard J.
N1 - Funding Information:
* Corresponding author. E-mail addresses: [email protected] (Y. Azar), [email protected] (L. Epstein), [email protected] (Y. Richter), [email protected] (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.
PY - 2004/8
Y1 - 2004/8
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=3042776074&partnerID=8YFLogxK
U2 - 10.1016/j.jalgor.2004.02.003
DO - 10.1016/j.jalgor.2004.02.003
M3 - Article
AN - SCOPUS:3042776074
SN - 0196-6774
VL - 52
SP - 120
EP - 133
JO - Journal of Algorithms
JF - Journal of Algorithms
IS - 2
ER -