Santa Claus Schedules Jobs on Unrelated Machines

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 the configuration LP, that was previously successfu...