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July  2012, 8(3): 651-656. doi: 10.3934/jimo.2012.8.651

## Worst-case performance of the successive approximation algorithm for four identical knapsacks

 1 Department of Mathematical Sciences, Tsinghua University, Beijing

Received  August 2011 Revised  February 2012 Published  June 2012

This paper studies the worst-case performance of the successive approximation algorithm for four identical knapsacks. The algorithm packs the knapsacks successively by using an exact algorithm on the remaining items for each single knapsack. We show that it is an $\frac{8}{11}$-approximation algorithm, and the bound is tight.
Citation: Zhenbo Wang. Worst-case performance of the successive approximation algorithm for four identical knapsacks. Journal of Industrial & Management Optimization, 2012, 8 (3) : 651-656. doi: 10.3934/jimo.2012.8.651
##### References:
 [1] A. Caprara and U. Pferschy, Worst-case analysis of the subset sum algorithm for bin packing,, Oper. Res. Lett., 32 (2004), 159.  doi: 10.1016/S0167-6377(03)00092-0.  Google Scholar [2] C. Chekuri and S. Khanna, A polynomial time approximation scheme for the multiple knapsack problem,, SIAM J. Comput., 35 (2005), 713.  doi: 10.1137/S0097539700382820.  Google Scholar [3] M. R. Garey and D. S. Johnson, "Computers and Intractability: A Guide to the Theory of NP-Completeness,", A Series of Books in the Mathematical Sciences, (1979).   Google Scholar [4] O. H. Ibarra and C. E. Kim, Fast approximation algorithms for the knapsack and sum of subset problem,, J. ACM, 22 (1975), 463.  doi: 10.1145/321906.321909.  Google Scholar [5] H. Kellerer, U. Pferschy and D. Pisinger, "Knapsack Problems,", Springer-Verlag, (2004).   Google Scholar [6] H. Kellerer and U. Pferschy, A new fully polynomial time approximation scheme for the knapsack problem,, J. Comb. Optim., 3 (1999), 59.   Google Scholar [7] D. Pisinger and P. Toth, Knapsack Problems,, in, (1998), 299.   Google Scholar [8] Z. Wang and W. Xing, A successive approximation algorithm for the multiple knapsack problem,, J. Comb. Optim., 17 (2009), 347.  doi: 10.1007/s10878-007-9116-y.  Google Scholar

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##### References:
 [1] A. Caprara and U. Pferschy, Worst-case analysis of the subset sum algorithm for bin packing,, Oper. Res. Lett., 32 (2004), 159.  doi: 10.1016/S0167-6377(03)00092-0.  Google Scholar [2] C. Chekuri and S. Khanna, A polynomial time approximation scheme for the multiple knapsack problem,, SIAM J. Comput., 35 (2005), 713.  doi: 10.1137/S0097539700382820.  Google Scholar [3] M. R. Garey and D. S. Johnson, "Computers and Intractability: A Guide to the Theory of NP-Completeness,", A Series of Books in the Mathematical Sciences, (1979).   Google Scholar [4] O. H. Ibarra and C. E. Kim, Fast approximation algorithms for the knapsack and sum of subset problem,, J. ACM, 22 (1975), 463.  doi: 10.1145/321906.321909.  Google Scholar [5] H. Kellerer, U. Pferschy and D. Pisinger, "Knapsack Problems,", Springer-Verlag, (2004).   Google Scholar [6] H. Kellerer and U. Pferschy, A new fully polynomial time approximation scheme for the knapsack problem,, J. Comb. Optim., 3 (1999), 59.   Google Scholar [7] D. Pisinger and P. Toth, Knapsack Problems,, in, (1998), 299.   Google Scholar [8] Z. Wang and W. Xing, A successive approximation algorithm for the multiple knapsack problem,, J. Comb. Optim., 17 (2009), 347.  doi: 10.1007/s10878-007-9116-y.  Google Scholar
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