Abstract
We consider the minimumnorm loadbalancing (MinNormLB) problem, wherein there are n jobs, each of which needs to be assigned to one of m machines, and we are given the processing times {p_{ij}} of the jobs on the machines. We also have a monotone, symmetric norm f:ℝ^m → ℝ_{≥ 0}. We seek an assignment σ of jobs to machines that minimizes the fnorm of the induced load vector load>_σ ∈ ℝ_{≥ 0}^m, where load_σ(i) = ∑_{j:σ(j) = i}p_{ij}. This problem was introduced by [Deeparnab Chakrabarty and Chaitanya Swamy, 2019], and the currentbest result for MinNormLB is a (4+ε)approximation [Deeparnab Chakrabarty and Chaitanya Swamy, 2019]. In the stochastic version of MinNormLB, the job processing times are given by nonnegative random variables X_{ij}, and jobs are independent; the goal is to find an assignment σ that minimizes the expected fnorm of the induced random load vector.
We obtain results that (essentially) match the bestknown guarantees for deterministic makespan minimization (MinNormLB with 𝓁_∞ norm). For MinNormLB, we obtain a (2+ε)approximation for unrelated machines, and a PTAS for identical machines. For stochastic MinNormLB, we consider the setting where the X_{ij}s are Poisson random variables, denoted PoisNormLB. Our main result here is a novel and powerful reduction showing that, for any machine environment (e.g., unrelated/identical machines), any αapproximation algorithm for MinNormLB in that machine environment yields a randomized α(1+ε)approximation for PoisNormLB in that machine environment. Combining this with our results for MinNormLB, we immediately obtain a (2+ε)approximation for PoisNormLB on unrelated machines, and a PTAS for PoisNormLB on identical machines. The latter result substantially generalizes a PTAS for makespan minimization with Poisson jobs obtained recently by [Anindya De et al., 2020].
BibTeX  Entry
@InProceedings{ibrahimpur_et_al:LIPIcs.ICALP.2021.81,
author = {Ibrahimpur, Sharat and Swamy, Chaitanya},
title = {{MinimumNorm Load Balancing Is (Almost) as Easy as Minimizing Makespan}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {81:181:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771955},
ISSN = {18688969},
year = {2021},
volume = {198},
editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14150},
URN = {urn:nbn:de:0030drops141504},
doi = {10.4230/LIPIcs.ICALP.2021.81},
annote = {Keywords: Approximation algorithms, Load balancing, Minimumnorm optimization, LP rounding}
}
Keywords: 

Approximation algorithms, Load balancing, Minimumnorm optimization, LP rounding 
Collection: 

48th International Colloquium on Automata, Languages, and Programming (ICALP 2021) 
Issue Date: 

2021 
Date of publication: 

02.07.2021 