Using dual approximation algorithms for scheduling problems theoretical and practical results
Journal of the ACM (JACM)
Tighter bounds on a heuristic for a partition problem
Information Processing Letters
Approximation algorithms for scheduling
Approximation algorithms for NP-hard problems
Approximation schemes for scheduling
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
An analysis of the LPT algorithm for the max-min and the min-ratio partition problems
Theoretical Computer Science
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This paper analyzes an approximation algorithm (Graham's LPT rule) for the NP-complete problem (Garey and Johnson, Computers and Intractability, Freeman, New York, 1979) of partitioning a given set of positive real numbers into k subsets. Chandra and Wong (SIAM J. Comput., 4(3) (1975) 249-263) analyzed the same algorithm for the L"2 metric for the similar classic scheduling problem and proved a 25/24 worst-case upper bound. This result was slightly improved by Leung and Wei (Inform. Process. Lett., 56 (1995) 51-57). This paper considers this algorithm on sets which have a k-partition with equal sum subsets and proves a new tight upper bound of 37/36 for the L"2 norm on such sets. To achieve this result we apply the statistical notion of variance to the partition obtained by the approximation algorithm. By Lagrange multiplier analysis, the maximum variance is shown not to exceed w"k"+"3/2 where w"k"+"3 is the k+3rd largest number in the set S. Sets of numbers are provided such that the tight bounds are achieved by the algorithm applied to this set.