# Scheduling to Approximate Minimization Objectives on Identical Machines

## TL;DR

This paper introduces a novel approximation algorithm for a generalized scheduling problem on multiple identical machines, achieving the first non-trivial results for objectives like throughput and tardiness using advanced linear programming techniques.

## Contribution

It presents the first $O( ext{log log } nP)$-approximation for scheduling with minimization objectives on multiple machines, utilizing strengthened linear programs and rounding methods.

## Key findings

- First non-trivial approximation for multiple machines with general objectives.
- Linear programming with job cover inequalities improves solution quality.
- Applicable to special cases like throughput minimization and tardiness.

## Abstract

This paper considers scheduling on identical machines. The scheduling objective considered in this paper generalizes most scheduling minimization problems. In the problem, there are $n$ jobs and each job $j$ is associated with a monotonically increasing function $g_j$. The goal is to design a schedule that minimizes $\sum_{j \in [n]} g_{j}(C_j)$ where $C_j$ is the completion time of job $j$ in the schedule. An $O(1)$-approximation is known for the single machine case. On multiple machines, this paper shows that if the scheduler is required to be either non-migratory or non-preemptive then any algorithm has an unbounded approximation ratio. Using preemption and migration, this paper gives a $O(\log \log nP)$-approximation on multiple machines, the first result on multiple machines. These results imply the first non-trivial positive results for several special cases of the problem considered, such as throughput minimization and tardiness. Natural linear programs known for the problem have a poor integrality gap. The results are obtained by strengthening a natural linear program for the problem with a set of covering inequalities we call job cover inequalities. This linear program is rounded to an integral solution by building on quasi-uniform sampling and rounding techniques.