Approximation ratio of LD algorithm for multi-processor scheduling and the Coffman–Sethi conjecture

Abstract

Coffman and Sethi proposed a heuristic algorithm, called LD (Longest Decreasing), for multi-processor scheduling, to minimize makespan over flowtime-optimal schedules. The LD algorithm is an extension of a very well-known list scheduling algorithm, Longest Processing Time (LPT) list scheduling, to this bicriteria scheduling problem. Coffman and Sethi conjectured (in 1976) that the LD algorithm has the following precise worst-case performance bound: 54−34(4m−1), where m is the number of machines. In this paper, utilizing some recent work by the authors and Huang (2016), which exposed some very strong combinatorial properties of various presumed minimal counterexamples to the conjecture, we provide a proof of this conjecture. The problem and the LD algorithm have connections to some other fundamental problems (such as the assembly line-balancing problem) and algorithms.