Abstract
We provide improved approximation algorithms for the min-max generalization problems considered by Du, Eppstein, Goodrich, and Lueker [1]. In min-max generalization problems, the input consists of data items with weights and a lower bound w lb, and the goal is to partition individual items into groups of weight at least w lb, while minimizing the maximum weight of a group. The rules of legal partitioning are specific to a problem. Du et al. consider several problems in this vein: (1) partitioning a graph into connected subgraphs, (2) partitioning unstructured data into arbitrary classes and (3) partitioning a 2-dimensional array into non-overlapping contiguous rectangles (subarrays) that satisfy the above size requirements.
We significantly improve approximation ratios for all the problems considered by Du et al., and provide additional motivation for these problems. Moreover, for the first problem, while Du et al. give approximation algorithms for specific graph families, namely, 3-connected and 4-connected planar graphs, no approximation algorithm that works for all graphs was known prior to this work.
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References
Du, W., Eppstein, D., Goodrich, M.T., Lueker, G.S.: On the approximability of geometric and geographic generalization and the min-max bin covering problem. In: WADS, pp. 242–253 (2009)
Ciriani, V., di Vimercati, S.D.C., Foresti, S., Samarati, P.: k-anonymous data mining: A survey. In: Aggarwal, C.C., Yu, P.S. (eds.) Privacy-Preserving Data Mining: Models and Algorithms, Springer, Heidelberg (2008)
Garcia, Y.J., Lopez, M.A., Leutenegger, S.T.: A greedy algorithm for bulk loading r-trees. In: GIS 1998: Proceedings of the 1998 ACM Int. Symp. on Advances in Geographic Information Systems, pp. 163–164. ACM, New York (1998)
Assmann, S.F., Johnson, D.S., Kleitman, D.J., Leung, J.Y.T.: On a dual version of the one-dimensional bin packing problem. J. Algorithms 5(4), 502–525 (1984)
Csirik, J., Johnson, D.S., Kenyon, C.: Better approximation algorithms for bin covering. In: SODA, pp. 557–566 (2001)
Jansen, K., Solis-Oba, R.: An asymptotic fully polynomial time approximation scheme for bin covering. Theor. Comput. Sci. 306(1-3), 543–551 (2003)
Bansal, N., Sviridenko, M.: The santa claus problem. In: STOC 2006: Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing, pp. 31–40. ACM, New York (2006)
Graham, R.L., Lawler, E.L., Lenstra, J.K., Kan, A.H.G.R.: Optimization and approximation in deterministic sequencing and scheduling: A survey. Annals of Discrete Mathematics 5, 287–326 (1979)
Manne, F.: Load Balancing in Parallel Sparse Matrix Computation. PhD thesis, University of Bergen, Norway (1993)
Khanna, S., Muthukrishnan, S., Paterson, M.: On approximating rectangle tiling and packing. In: SODA, pp. 384–393 (1998)
Sharp, J.P.: Tiling multi-dimensional arrays. In: Ciobanu, G., Păun, G. (eds.) FCT 1999. LNCS, vol. 1684, pp. 500–511. Springer, Heidelberg (1999)
Smith, A., Suri, S.: Rectangular tiling in multi-dimensional arrays. In: SODA, pp. 786–794 (1999)
Muthukrishnan, S., Poosala, V., Suel, T.: On rectangular partitionings in two dimensions: Algorithms, complexity, and applications. In: Beeri, C., Bruneman, P. (eds.) ICDT 1999. LNCS, vol. 1540, pp. 236–256. Springer, Heidelberg (1998)
Berman, P., DasGupta, B., Muthukrishnan, S., Ramaswami, S.: Improved approximation algorithms for rectangle tiling and packing. In: SODA, pp. 427–436 (2001)
Berman, P., DasGupta, B., Muthukrishnan, S.: Slice and dice: A simple, improved approximate tiling recipe. In: SODA, pp. 455–464 (2002)
Berman, P., DasGupta, B., Muthukrishnan, S.: Approximation algorithms for max-min tiling. J. Algorithms 47(2), 122–134 (2003)
Lenstra, J.K., Shmoys, D.B., Tardos, E.: Approximation algorithms for scheduling unrelated parallel machines. Math. Program. 46(3), 259–271 (1990)
Tutte, W.T.: A theorem on planar graphs. Trans. Amer. Math. Soc. 82, 99–116 (1956)
Chiba, N., Nishizeki, T.: The hamiltonian cycle problem is linear-time solvable for 4-connected planar graphs. J. Algorithms 10(2), 187–211 (1989)
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Berman, P., Raskhodnikova, S. (2010). Approximation Algorithms for Min-Max Generalization Problems. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. RANDOM APPROX 2010 2010. Lecture Notes in Computer Science, vol 6302. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15369-3_5
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DOI: https://doi.org/10.1007/978-3-642-15369-3_5
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