Speed-robust scheduling revisited

TL;DR

This paper improves bounds on speed-robust scheduling for related machines, providing tighter guarantees for equal and small jobs, and introduces a new case with near-optimal robustness.

Contribution

It presents an improved 1.6-robust bound for equal-size jobs with b=m, tight bounds for infinitesimal jobs when b≥m, and a new case with near 2-robustness for small jobs.

Findings

Improved robustness bound of 1.6 for equal-size jobs with b=m.

Tight bounds established for infinitesimal jobs when b≥m.

New algorithm achieves near 2-robustness for a class of small jobs.

Abstract

Speed-robust scheduling is the following two-stage problem of scheduling $n$ jobs on $m$ uniformly related machines. In the first stage, the algorithm receives the value of $m$ and the processing times of $n$ jobs; it has to partition the jobs into $b$ groups called bags. In the second stage, the machine speeds are revealed and the bags are assigned to the machines, i.e., the algorithm produces a schedule where all the jobs in the same bag are assigned to the same machine. The objective is to minimize the makespan (the length of the schedule). The algorithm is compared to the optimal schedule and it is called $\rho$-robust, if its makespan is always at most $\rho$ times the optimal one. Our main result is an improved bound for equal-size jobs for $b=m$. We give an upper bound of $1.6$. This improves previous bound of $1.8$ and it is almost tight in the light of previous lower bound of $1.58$. Second, for infinitesimally small jobs, we give tight upper and lower bounds for the case when $b\geq m$. This generalizes and simplifies the previous bounds for $b=m$. Finally, we introduce a new special case with relatively small jobs for which we give an algorithm whose robustness is close to that of infinitesimal jobs and thus gives better than $2$-robust for a large class of inputs.