Abstract
We study the basic problem of assigning memoryless workers to tasks with dynamically changing demands. Given a set of w workers and a multiset T ⊆ [t] of T = w tasks, a memoryless workertask assignment function is any function ϕ that assigns the workers [w] to the tasks T based only on the current value of T. The assignment function ϕ is said to have switching cost at most k if, for every task multiset T, changing the contents of T by one task changes ϕ(T) by at most k worker assignments. The goal of memoryless worker task assignment is to construct an assignment function with the smallest possible switching cost.
In past work, the problem of determining the optimal switching cost has been posed as an open question. There are no known sublinear upper bounds, and after considerable effort, the best known lower bound remains 4 (ICALP 2020).
We show that it is possible to achieve polylogarithmic switching cost. We give a construction via the probabilistic method that achieves switching cost O(log w log (wt)) and an explicit construction that achieves switching cost polylog (wt). We also prove a superconstant lower bound on switching cost: we show that for any value of w, there exists a value of t for which the optimal switching cost is w. Thus it is not possible to achieve a switching cost that is sublinear strictly as a function of w.
Finally, we present an application of the workertask assignment problem to a metric embeddings problem. In particular, we use our results to give the first lowdistortion embedding from sparse binary vectors into lowdimensional Hamming space.
BibTeX  Entry
@InProceedings{berger_et_al:LIPIcs.ICALP.2022.19,
author = {Berger, Aaron and Kuszmaul, William and Polak, Adam and Tidor, Jonathan and Wein, Nicole},
title = {{Memoryless WorkerTask Assignment with Polylogarithmic Switching Cost}},
booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
pages = {19:119:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772358},
ISSN = {18688969},
year = {2022},
volume = {229},
editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16360},
URN = {urn:nbn:de:0030drops163608},
doi = {10.4230/LIPIcs.ICALP.2022.19},
annote = {Keywords: Distributed Task Allocation, Metric Embeddings, Probabilistic Method}
}
Keywords: 

Distributed Task Allocation, Metric Embeddings, Probabilistic Method 
Collection: 

49th International Colloquium on Automata, Languages, and Programming (ICALP 2022) 
Issue Date: 

2022 
Date of publication: 

28.06.2022 