Santa Claus Schedules Jobs on Unrelated Machines

TL;DR

This paper presents a new polynomial-time algorithm for the restricted assignment scheduling problem, achieving an approximation ratio of approximately 1.9412, improving upon previous bounds by analyzing the configuration LP and using local search.

Contribution

Introduces a polynomial-time algorithm with a 1.9412 approximation ratio for the restricted assignment problem, surpassing previous approximation bounds.

Findings

Achieves a 1.9412 approximation ratio for the problem.

Bounds the integrality gap of the configuration LP.

Uses local search to find near-optimal schedules.

Abstract

One of the classic results in scheduling theory is the 2-approximation algorithm by Lenstra, Shmoys, and Tardos for the problem of scheduling jobs to minimize makespan on unrelated machines, i.e., job j requires time p_{ij} if processed on machine i. More than two decades after its introduction it is still the algorithm of choice even in the restricted model where processing times are of the form p_{ij} in {p_j, \infty}. This problem, also known as the restricted assignment problem, is NP-hard to approximate within a factor less than 1.5 which is also the best known lower bound for the general version. Our main result is a polynomial time algorithm that estimates the optimal makespan of the restricted assignment problem within a factor 33/17 + \epsilon \approx 1.9412 + \epsilon, where \epsilon > 0 is an arbitrarily small constant. The result is obtained by upper bounding the integrality gap of a certain strong linear program, known as configuration LP, that was previously successfully used for the related Santa Claus problem. Similar to the strongest analysis for that problem our proof is based on a local search algorithm that will eventually find a schedule of the mentioned approximation guarantee, but is not known to converge in polynomial time.