Abstract
This paper analyzes the NP-complete problem (M.R. Garey, D.S. Johnson, Computers and Intractability, Freeman, New York, 1979) of finding an optimal partition of k subsets for a given set of positive real numbers. A number of norms ( L 2, L ∞ and more generally, L p ) have been utilized to measure the competitive ratio of the partition obtained. Chandra and Wong (A.K. Chandra, C.K. Wong, Worst-case analysis of a placement algorithm related to storage allocation, SIAM Journal on Computing 4 (3) (1975) 249–264) analyzed Graham's LPT rule for the L p norm and proved a 3/2 worst case upper bound in the general case. A class of algorithms to solve this partitioning problem is introduced that relaxes Graham's LPT rule (R.L. Graham, Bounds on multiprocessing timing anomalies, SIAM Journal of Applied Mathematics 17 (1969) 263–269) so that the current element can be placed in (almost) any subset instead of just the minimal subset. The input sets considered in this paper are sets which can yield a k-partition with equal sum subsets (termed ideal sets). Specifically, it is shown that for the L p norm, any algorithm of the class has an upper bound of 4/3 for the competitive ratio on ideal sets. It is also shown that this is the best possible such upper bound.
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