Worst-case analysis of the differencing method for the partition problem
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Discrete Applied Mathematics
The differencing algorithm LDM for partitioning: a proof of a conjecture of Karmarkar and Karp
Mathematics of Operations Research
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Artificial Intelligence
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IJCAI'11 Proceedings of the Twenty-Second international joint conference on Artificial Intelligence - Volume Volume One
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We consider the problem of partitioning a set of n numbers into m subsets of cardinality k = 驴n/m驴 or 驴n/m驴, such that the maximum subset sum is minimal. We prove that the performance ratios of the Differencing Method of Karmarkar and Karp for k = 3,4,5, and 6 are precisely 4/3, 19/12, 103/60, and 643/360, respectively, by means of a novel approach in which the ratios are explicitly calculated using mixed integer linear programming. Moreover, we show that for k 驴 7 the performance ratio lies between 2-2/k and 2-1/(k-1). For the case that m is given instead of k, we prove a performance ratio of precisely 2-1/m. The results settle the problem of determining theworst-case performance of the Differencing Method.