Approximation Ratio of LD Algorithm for Multi-Processor Scheduling and the Coffman-Sethi Conjecture

TL;DR

This paper proves the Coffman-Sethi conjecture, establishing the exact worst-case performance bound of the LD heuristic algorithm for multi-processor scheduling, which is crucial for optimizing makespan in flowtime-optimal schedules.

Contribution

The paper provides a proof of the long-standing Coffman-Sethi conjecture regarding the approximation ratio of the LD algorithm for multi-processor scheduling.

Findings

Confirmed the conjectured worst-case performance bound of LD algorithm

Established strong combinatorial properties of minimal counterexamples

Connected the problem to other fundamental scheduling problems

Abstract

Coffman and Sethi proposed a heuristic algorithm, called LD, for multi-processor scheduling, to minimize makespan over flowtime-optimal schedules. LD algorithm is a natural extension of a very well-known list scheduling algorithm, Longest Processing Time (LPT) list scheduling, to our bicriteria scheduling problem. Moreover, in 1976, Coffman and Sethi conjectured that LD algorithm has precisely the following worst-case performance bound: $\frac{5}{4} - \frac{3}{4(4m-1)}$, where m is the number of machines. In this paper, utilizing some recent work by the authors and Huang, from 2013, 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 other fundamental problems (such as the assembly line-balancing problem) and to other algorithms.