Performance Ratios for the Differencing Method Applied to the Balanced Number Partitioning Problem

We consider the problem of partitioning a set of n numbers into m subsets of cardinality k = ?n/m? or ?n/m?, 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 ? 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.

[1]  E. G. Coffman,et al.  A Note on Expected Makespans for Largest-First Sequences of Independent Tasks on Two Processors , 1984, Math. Oper. Res..

[2]  Hans Kellerer,et al.  A 7/6–Approximation Algorithm For 3-Partitioning And Its Application To Multiprocessor Scheduling , 1999 .

[3]  Gerhard J. Woeginger,et al.  A Tight Bound for 3-Partitioning , 1993, Discret. Appl. Math..

[4]  Benjamin Yakir,et al.  The Differencing Algorithm LDM for Partitioning: A Proof of a Conjecture of Karmarkar and Karp , 1996, Math. Oper. Res..

[5]  Jan van Leeuwen,et al.  Performance Ratios for the Karmarkar-Karp Differencing Method , 2003, Electron. Notes Discret. Math..

[6]  Li-Hui Tsai.  The loading and scheduling problems in flexible manufacturing systems , 1987 .

[7]  Li-Hui Tsai,et al.  The Modified Differencing Method for the Set Partitioning Problem with Cardinality Constraints , 1995, Discret. Appl. Math..

[8]  Ronald L. Graham,et al.  Bounds for certain multiprocessing anomalies , 1966 .

[9]  Michael O. Ball,et al.  Sequencing of Insertions in Printed Circuit Board Assembly , 1988, Oper. Res..

[10]  Wil Michiels,et al.  Min–max subsequence problems in multi‐zone disk recording , 2001 .

[11]  Richard M. Karp,et al.  The Differencing Method of Set Partitioning , 1983 .

[12]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[13]  Edward G. Coffman,et al.  An Application of Bin-Packing to Multiprocessor Scheduling , 1978, SIAM J. Comput..

[14]  Li-Hui Tsai,et al.  Asymptotic Analysis of an Algorithm for Balanced Parallel Processor Scheduling , 1992, SIAM J. Comput..

[15]  Stephan Mertens.  A complete anytime algorithm for balanced number partitioning , 1999, ArXiv.

[16]  Richard E. Korf,et al.  A Complete Anytime Algorithm for Number Partitioning , 1998, Artif. Intell..

[17]  Ronald L. Graham,et al.  Bounds on Multiprocessing Timing Anomalies , 1969, SIAM Journal of Applied Mathematics.

[18]  Matteo Fischetti,et al.  Worst-case analysis of the differencing method for the partition problem , 1987, Math. Program..

[19]  George S. Lueker.  A note on the average-case behavior of a simple differencing method for partitioning , 1987 .

[20]  Ward WhittAT,et al.  Recent Asymptotic Results in the Probabilistic Analysis of Schedule Makespans Recent Asymptotic Results in the Probabilistic Analysis of Schedule Makespans , 1995 .

[21]  Richard C. Larson,et al.  Model Building in Mathematical Programming , 1979 .