# A Quasi-Polynomial Approximation for the Restricted Assignment Problem

## TL;DR

This paper improves the approximation ratio for the Restricted Assignment Problem using a quasi-polynomial algorithm, reducing the integrality gap and simplifying the proof of the LP relaxation.

## Contribution

It provides the first quasi-polynomial time algorithm with an approximation ratio better than 2 for the problem, improving previous bounds and simplifying analysis.

## Key findings

- Integrality gap improved from 1.9142 to 1.8334
- First quasi-polynomial approximation algorithm for the problem
- Simplified proof of the LP relaxation bound

## Abstract

The Restricted Assignment Problem is a prominent special case of Scheduling on Parallel Unrelated Machines. For the strongest known linear programming relaxation, the configuration LP, we improve the non-constructive bound on its integrality gap from 1.9142 to 1.8334 and significantly simplify the proof. Then we give a constructive variant, yielding a 1.8334-approximation in quasi-polynomial time. This is the first quasi-polynomial algorithm for this problem improving on the long-standing approximation rate of 2.