Worst-case performance analysis of some approximation algorithms for minimizing makespan and flowtime

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 the LD algorithm, has a simple worst‐case performance bound: $\frac{5m‐2}{4m‐1}$ , where m is the number of machines. We study the structure of potential minimal counterexamples to this conjecture, provide some new tools and techniques for the analysis of such algorithms, and prove that to verify the conjecture, it suffices to analyze the following case: for every $m\ge 4$ , $n\in \{4m,5m\}$ , where n is the number of jobs.