Document

Optimal Matroid Partitioning Problems

File

LIPIcs.ISAAC.2017.51.pdf
• Filesize: 0.57 MB
• 13 pages

Cite As

Yasushi Kawase, Kei Kimura, Kazuhisa Makino, and Hanna Sumita. Optimal Matroid Partitioning Problems. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 51:1-51:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.ISAAC.2017.51

Abstract

This paper studies optimal matroid partitioning problems for various objective functions. In the problem, we are given a finite set E and k weighted matroids (E, \mathcal{I}_i, w_i), i = 1, \dots, k, and our task is to find a minimum partition (I_1,\dots,I_k) of E such that I_i \in \mathcal{I}_i for all i. For each objective function, we give a polynomial-time algorithm or prove NP-hardness. In particular, for the case when the given weighted matroids are identical and the objective function is the sum of the maximum weight in each set (i.e., \sum_{i=1}^k\max_{e\in I_i}w_i(e)), we show that the problem is strongly NP-hard but admits a PTAS.
Keywords
• Matroids
• Partitioning problem
• PTAS
• NP-hardness

Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

References

1. L. Babel, H. Kellerer, and V. Kotov. The k-partitioning problem. Mathematical Methods of Operations Research, 47:59-82, 1998.
2. R. E. Burkard and E. Yao. Constrained partitioning problems. Discrete Applied Mathematics, 28:21-34, 1990.
3. S.-P. Chen, Y. He, and G. Lin. 3-partitioning for maximizing the minimum load. Journal of Combinatorial Optimization, 6:67-80, 2002.
4. M. Dell'Amico and S. Martello. Bounds for the cardinality constrained p||c_max problem. Journal of Scheduling, 4:123-138, 2001.
5. P. Dell'Olmo, P. Hansen, S. Pallottino, and G. Storchi. On uniform k-partition problems. Discrete Applied Mathematics, 150:121-139, 2005.
6. I. Dinur and D. Steurer. Analytical approach to parallel repetition. In Proceedings of the Forty-sixth Annual ACM Symposium on Theory of Computing, pages 624-633. ACM, 2014.
7. J. Edmonds. Minimum partition of a matroid into independent subsets. JOURNAL OF RESEARCH of the National Bureau of Standards - B. Mathematicsnd Mathematical Physics, 69B:67-72, 1965.
8. J. Edmonds. Submodular functions, matroids, and certain polyhedra. In Combinatorial Structures and their Applications, pages 69-87. Gordon and Breach, New York, 1970.
9. J. Edmonds and D. R. Fulkerson. Transversals and matroid partition. Journal of Research of the National Bureau of Standards, 69B:147-153, 1965.
10. U. Feige, D. Peleg, and G. Kortsarz. The dense k-subgraph problem. Algorithmica, 29(3):410-421, 2001.
11. A. Frank. A weighted matroid intersection algorithm. Journal of Algorithms, 2(4):328-336, 1981.
12. M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman New York, 1979.
13. Y. He, Z. Tan, J. Zhu, and E. Yao. k-partitioning problems for maximizing the minimum load. Computers and Mathematics with Applications, 46:1671-1681, 2003.
14. Y. Kawase, K. Kimura, K. Makino, and H. Sumita. Optimal Matroid Partitioning Problems. arXiv preprints cs.DS/1710.00950, October 2017. URL: http://arxiv.org/abs/1710.00950.
15. H. Kellerer and G. Woeginger. A tight bound for 3-partitioning. Discrete Applied Mathematics, 45:249-259, 1993.
16. B. Korte and J. Vygen. Combinatorial Optimization: Theory and Algorithms. Springer, 2002.
17. E. L. Lawler. Matroids with parity conditions: A new class of combinatorial optimization problems. Memo erl-m334, Electronics Research Laboratory, College of Engineering, UC Berkeley, Berkeley, CA, 1971.
18. J. Lee, M. Sviridenko, and J. Vondrák. Matroid matching: The power of local search. SIAM Journal on Computing, 42(1):357-379, 2013.
19. J. K. Lenstra, D. B. Shmoys, and É. Tardos. Approximation algorithms for scheduling unrelated parallel machines. Mathematical Programming, 46:259-271, 1990.
20. W. Li and J. Li. Approximation algorithms for k-partitioning problems with partition matroid constraint. Optimization Letters, 8:1093-1099, 2014.
21. L. Lovász. Matroid matching and some applications. Journal of Combinatorial Theory, Series B, 28(2):208-236, 1980.
22. D. Moshkovitz. The projection games conjecture and the NP-hardness of ln n-approximating set-cover. Theory of Computing, 11(7):221-235, 2015.
23. A. Schrijver. Combinatorial Optimization. Springer, 2003.
24. J. Verschae and A. Wiese. On the configuration-LP for scheduling on unrelated machines. Journal of Scheduling, 17:371-383, 2014.
25. B. Wu and E. Yao. k-partitioning problems with partition matroid constraint. Theoretical Computer Science, 374:41-48, 2007.
26. B. Wu and E. Yao. Lower bounds and modified LPT algorithm for k-partitioning problems with partition matroid constraint. Applied Mathematics-A Journal of Chinese Universities, 23:1-8, 2008.
X

Feedback for Dagstuhl Publishing