Abstract

We consider a special case of the scheduling problem on unrelated machines, namely the restricted assignment problem with two different processing times. We show that the configuration LP has an integrality gap of at most $$\frac{5}{3} \approx 1.667$$ for this problem. This allows us to estimate the optimal makespan within a factor of $$\frac{5}{3}$$ , improving upon the previously best known estimation algorithm with ratio $$\frac{11}{6} \approx 1.833$$ due to Chakrabarty et al. (in: Proceedings of the twenty-sixth annual ACM-SIAM symposium on discrete algorithms (SODA 2015), pp 1087–1101, 2015).

Highlights

  • Scheduling on unrelated machines is a problem where we are given a set J of jobs and a set M of machines, and the processing time of job j ∈ J on machine i ∈ M is given by pij

  • The algorithm uses a rounding procedure for the following, natural linear programming formulation, which is commonly known as the assignment linear program (LP): xij = 1 for each j ∈ J

  • We present better bounds on the integrality gap of the configuration LP

Read more

Summary

Introduction

Scheduling on unrelated machines is a problem where we are given a set J of jobs and a set M of machines, and the processing time of job j ∈ J on machine i ∈ M is given by pij. Min{T | CLP(T ) has a feasible integer solution} and OPTLP(I) analogously With this definition, OPT(I) is equal to the makespan of an optimal schedule. Note that if OPTLP ≥ 2b, the analysis of Lenstra, Shmoys, and Tardos [6] bounds the integrality gap of the assignment LP by 1.5, and the configuration LP is at least as strong. Found a the general case and improved the estimation ratio to 1.833 They presented a constructive algorithm with approximation ratio 2 − δ for a very small δ > 0. Our algorithm has an additive approximation guarantee of OPTLP + b − s, which might be of independent interest

Bounding the Integrality Gap by Local Search
Detailed Description of the Algorithm
Proof of Termination
Improving the Bound by Scaling
Case 1
Case 2

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.