Towards Tight Lower Bounds for Scheduling Problems

TL;DR

This paper establishes tight inapproximability bounds for various scheduling problems with precedence constraints by connecting them to structural hardness in $k$-partite graphs, assuming a generalized hardness hypothesis.

Contribution

It introduces a novel connection between $k$-partite graph hardness and scheduling inapproximability, achieving tight bounds and resolving open questions.

Findings

Hardness of $2-$ for makespan minimization on precedence-constrained jobs.

Super constant inapproximability for scheduling on related parallel machines.

Unification of inapproximability results for multiple scheduling problems.

Abstract

We show a close connection between structural hardness for $k$-partite graphs and tight inapproximability results for scheduling problems with precedence constraints. Assuming a natural but nontrivial generalisation of the bipartite structural hardness result of Bansal and Khot, we obtain a hardness of $2-\epsilon$ for the problem of minimising the makespan for scheduling precedence-constrained jobs with preemption on identical parallel machines. This matches the best approximation guarantee for this problem. Assuming the same hypothesis, we also obtain a super constant inapproximability result for the problem of scheduling precedence-constrained jobs on related parallel machines, making progress towards settling an open question in both lists of ten open questions by Williamson and Shmoys, and by Schuurman and Woeginger. The study of structural hardness of $k$-partite graphs is of independent interest, as it captures the intrinsic hardness for a large family of scheduling problems. Other than the ones already mentioned, this generalisation also implies tight inapproximability to the problem of minimising the weighted completion time for precedence-constrained jobs on a single machine, and the problem of minimising the makespan of precedence-constrained jobs on identical parallel machine, and hence unifying the results of Bansal and Khot, and Svensson, respectively.