Set cover in sub-linear time

Piotr Indyk, Sepideh Mahabadi, Ronitt Rubinfeld, Ali Vakilian, Anak Yodpinyanee

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

8 Scopus citations

Abstract

We study the classic set cover problem from the perspective of sub-linear algorithms. Given access to a collection of m sets over n elements in the query model, we show that sub-linear algorithms derived from existing techniques have almost tight query complexities. On one hand, first we show an adaptation of the streaming algorithm presented in [17] to the sub-linear query model, that returns an θ-approximate cover using eO (m(n=k)1=(1) + nk) queries to the input, where k denotes the value of a minimum set cover. We then complement this upper bound by proving that for lower values of k, the required number of queries is e(m(n=k)1=(2)), even for estimating the optimal cover size. Moreover, we prove that even checking whether a given collection of sets covers all the elements would require (nk) queries. These two lower bounds provide strong evidence that the upper bound is almost tight for certain values of the parameter k. On the other hand, we show that this bound is not optimal for larger values of the parameter k, as there exists a (1+ϵ)-approximation algorithm with O(mn=kϵ2) queries. We show that this bound is essentially tight for sufficiently small constant ", by establishing a lower bound of e(mn=k) query complexity. Our lower-bound results follow by carefully designing two distributions of instances that are hard to distinguish. In particular, our first lower bound involves a probabilistic construction of a certain set system with a minimum set cover of size k, with the key property that a small number of "almost uniformly distributed" modifications can reduce the minimum set cover size down to k. Thus, these modifications are not detectable unless a large number of queries are asked. We believe that our probabilistic construction technique might find applications to lower bounds for other combinatorial optimization problems.

Original languageEnglish
Title of host publication29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018
EditorsArtur Czumaj
PublisherAssociation for Computing Machinery
Pages2467-2486
Number of pages20
ISBN (Electronic)9781611975031
DOIs
StatePublished - 2018
Externally publishedYes
Event29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 - New Orleans, United States
Duration: 7 Jan 201810 Jan 2018

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Conference

Conference29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018
Country/TerritoryUnited States
CityNew Orleans
Period7/01/1810/01/18

Funding

FundersFunder number
National Science Foundation
Directorate for Computer and Information Science and Engineering1733808, 1420692, 1741137

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