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Worst-Case Performance Analysis of Some Approximation Algorithms for Minimizing Makespan and Flow-Time

In 1976, Coffman and Sethi conjectured that a natural extension of LPT list scheduling to the bicriteria scheduling problem of minimizing makespan over flowtime optimal schedules, called LD algorithm, has a simple worst-case performance bound: (5m-2)/(4m-1), where m is the number of machines. We study structure of potential minimal counterexamples to this conjecture and prove that the conjecture holds for the cases (i) n > 5m, (ii) m = 2, (iii) m = 3, and (iv) m greater than or equal to 4, n less than or equal to 3m, where n is the number of jobs. We further conclude that to verify the conjecture, it suffices to analyze the following case: for every m greater than or equal to 4, n is either equal to 4m or 5m.