# Non-uniform Geometric Set Cover and Scheduling on Multiple Machines

## TL;DR

This paper presents an $O(1)$ approximation algorithm for a complex scheduling problem with non-decreasing cost functions, improving previous bounds and introducing a novel geometric approach to handle non-uniform capacities.

## Contribution

The authors develop a new method to preserve geometric structure in capacitated covering problems, enabling constant-factor approximations for scheduling and related problems.

## Key findings

- Achieved $O(1)$ approximation for the scheduling problem.
- Introduced a geometric technique to handle non-uniform capacities.
- Extended the approach to other capacitated covering problems.

## Abstract

We consider the following general scheduling problem studied recently by Moseley. There are $n$ jobs, all released at time $0$, where job $j$ has size $p_j$ and an associated arbitrary non-decreasing cost function $f_j$ of its completion time. The goal is to find a schedule on $m$ machines with minimum total cost. We give an $O(1)$ approximation for the problem, improving upon the previous $O(\log \log nP)$ bound ($P$ is the maximum to minimum size ratio), and resolving the open question of Moseley. We first note that the scheduling problem can be reduced to a clean geometric set cover problem where points on a line with arbitrary demands, must be covered by a minimum cost collection of given intervals with non-uniform capacity profiles. Unfortunately, current techniques for such problems based on knapsack cover inequalities and low union complexity, completely lose the geometric structure in the non-uniform capacity profiles and incur at least an $\Omega(\log\log P)$ loss. To this end, we consider general covering problems with non-uniform capacities, and give a new method to handle capacities in a way that completely preserves their geometric structure. This allows us to use sophisticated geometric ideas in a black-box way to avoid the $\Omega(\log \log P)$ loss in previous approaches. In addition to the scheduling problem above, we use this approach to obtain $O(1)$ or inverse Ackermann type bounds for several basic capacitated covering problems.