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Approximation algorithms for \(k\)-partitioning problems with partition matroid constraint

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Abstract

The \(k\)-partitioning problem with partition matroid constraint is to partition the union of \(k\) given sets of size \(m\) into \(m\) sets such that each set contains exactly one element from each given set. With the objective of minimizing the maximum load, we present an efficient polynomial time approximation scheme (EPTAS) for the case where \(k\) is a constant and a full polynomial time approximation scheme (FPTAS) for the case where \(m\) is a constant; with the objective of maximizing the minimum load, we present a \(\frac{1}{k-1}\)-approximation algorithm for the general case, an EPTAS for the case where \(k\) is a constant; with the objective of minimizing the \(l_p\)-norm of the load vector, we prove that the layered LPT algorithm (Wu and Yao in Theor Comput Sci 374:41–48, 2007) is an all-norm 2-approximation algorithm.

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References

  1. Babel, L., Kellerer, H., Kotov, V.: The \(k\)-partitioning problem. Math. Methods Oper. Res. 47, 59–82 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bruglieri, M., Ehrgott, M., Hamacher, H.W., Maffioli, F.: An annotated bibliography of combinatorial optimization problems with fixed cardinality constraints. Discrete Appl. Math. 154, 1344–1357 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chen, S., He, Y., Lin, G.: 3-partitioning for maximizing the minimum load. J. Comb. Optim. 6, 67–80 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Dell’Amico, M., Iori, M., Martello, S.: Heuristic algorithms and scatter search for the cardinality constrained \(P|| C_{max}\) problem. J. Heuristics 10, 169–204 (2004)

    Article  Google Scholar 

  5. Dell’Amico, M., Iori, M., Martello, S., Monaci, M.: Lower bound and heuristic algorithms for the \(k_i\) partitioning problem. Eur. J. Oper. Res. 171, 725–742 (2006)

    Google Scholar 

  6. Dell’Amico, M., Martello, S.: Bounds for the cardinality constrained \(P||C_{max}\) problem. J. Sched. 4, 123–138 (2001)

    Google Scholar 

  7. Dell’ Olmo, P., Hansen, P., Pallottino, S., Storchi, G.: On uniform \(k\)-partition problems. Discrete Appl. Math. 150, 121–139 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. He, Y., Tan, Z., Zhu, J., Yao, E.: \(k\)-Partitioning problems for maximizing the minimum load. Comput. Math. Appl. 46, 1671–1681 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kellerer, H., Kotov, V.: A \(\frac{7}{6}\)-approximation algorithm for 3-partitioning and its application to multiprocessor scheduling. INFOR 37, 48–56 (1999)

    Google Scholar 

  10. Kellerer, H., Woeginger, G.: A tight bound for 3-partitioning. Discrete Appl. Math. 45, 249–259 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  11. Lenstra, H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8, 538–548 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  12. Wu, B., Yao, E.: \(k\)-Partitioning problems with partition matroid constraint. Theor. Comput. Sci. 374, 41–48 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Wu, B., Yao, E.: Lower bounds and modified LPT algorithm for \(k\)-partitioning problems with partition matroid constraint. Appl. Math. J. Chin. Univ. 23, 1–8 (2008)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The authors are grateful to the two anonymous referees whose comments and suggestions have led to a substantially improved presentation for the paper. The work is supported by the Tianyuan Fund for Mathematics of the National Natural Science Foundation of China [No. 11126315] and the National Natural Science Foundation of China [No. 61063011].

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Authors and Affiliations

  1. Department of Atmospheric Science, Yunnan University, 650091 , Kunming, People’s Republic of China

    Weidong Li

  2. Department of Mathematics, Yunnan University, 650091 , Kunming, People’s Republic of China

    Jianping Li

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  1. Weidong Li
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Correspondence to Weidong Li.

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Li, W., Li, J. Approximation algorithms for \(k\)-partitioning problems with partition matroid constraint. Optim Lett 8, 1093–1099 (2014). https://doi.org/10.1007/s11590-013-0637-2

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  • DOI: https://doi.org/10.1007/s11590-013-0637-2

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