On the Configuration-LP of the Restricted Assignment Problem

TL;DR

This paper improves the upper bound on the integrality gap of the configuration-LP for the Restricted Assignment problem from approximately 1.9412 to 1.8333, simplifying the proof and enabling better approximation algorithms.

Contribution

It simplifies Svensson's proof and tightens the integrality gap bound for the configuration-LP in the Restricted Assignment problem.

Findings

The integrality gap bound is improved to 2 - 1/6 (~1.8333).

A polynomial-time approximation algorithm with ratio 2 - 1/6 + ε is achieved.

The proof simplification makes the bound more accessible and easier to analyze.

Abstract

We consider the classical problem of Scheduling on Unrelated Machines. In this problem a set of jobs is to be distributed among a set of machines and the maximum load (makespan) is to be minimized. The processing time $p_{ij}$ of a job $j$ depends on the machine $i$ it is assigned to. Lenstra, Shmoys and Tardos gave a polynomial time $2$-approximation for this problem. In this paper we focus on a prominent special case, the Restricted Assignment problem, in which $p_{ij}\in\{p_j,\infty\}$. The configuration-LP is a linear programming relaxation for the Restricted Assignment problem. It was shown by Svensson that the multiplicative gap between integral and fractional solution, the integrality gap, is at most $2 - 1/17 \approx 1.9412$. In this paper we significantly simplify his proof and achieve a bound of $2 - 1/6 \approx 1.8333$. As a direct consequence this provides a polynomial $(2 - 1/6 + \epsilon)$-estimation algorithm for the Restricted Assignment problem by approximating the configuration-LP. The best lower bound known for the integrality gap is $1.5$ and no estimation algorithm with a guarantee better than $1.5$ exists unless $\mathrm{P} = \mathrm{NP}$.