Affiliation:
1. Technion—Israel Institute of Technology, Haifa, Israel
2. CNRS and Université Paris Diderot, France
Abstract
This article studies the set cover problem under the semi-streaming model. The underlying set system is formalized in terms of a hypergraph
G
= (
V
,
E
) whose edges arrive one by one, and the goal is to construct an edge cover
F
⊆
E
with the objective of minimizing the cardinality (or cost in the weighted case) of
F
. We further consider a parameterized relaxation of this problem, where, given some 0 ⩽ ϵ < 1, the goal is to construct an edge (1 − ϵ)-cover, namely, a subset of edges incident to all but an ϵ-fraction of the vertices (or their benefit in the weighted case). The key limitation imposed on the algorithm is that its space is limited to (poly)logarithmically many bits per vertex.
Our main result is an asymptotically tight tradeoff between ϵ and the approximation ratio: We design a semi-streaming algorithm that on input hypergraph
G
constructs a succinct data structure
D
such that for every 0 ⩽ ϵ < 1, an edge (1 − ϵ)-cover that approximates the optimal edge (1-)cover within a factor of
f
(ϵ,
n
) can be extracted from
D
(efficiently and with no additional space requirements), where
f
(ϵ,
n
) = {
O
(1/
ϵ
), if ϵ > 1/√
n
O
(√
n
), otherwise
. In particular, for the traditional set cover problem, we obtain an
O
(√
n
-approximation. This algorithm is proved to be best possible by establishing a family (parameterized by ϵ) of matching lower bounds.
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Reference34 articles.
1. K. J. Ahn and S. Guha. 2009. Graph sparsification in the semi-streaming model. In ICALP. 328--338. 10.1007/978-3-642-02930-1_27 K. J. Ahn and S. Guha. 2009. Graph sparsification in the semi-streaming model. In ICALP. 328--338. 10.1007/978-3-642-02930-1_27
2. The Online Set Cover Problem
3. Adversarial Leakage in Games
4. Streaming algorithm for graph spanners—single pass and constant processing time per edge
Cited by
4 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献
1. Set Cover in the One-pass Edge-arrival Streaming Model;Proceedings of the 42nd ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems;2023-06-18
2. Random Order Online Set Cover is as Easy as Offline;2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS);2022-02
3. Online coverage and inspection planning for 3D modeling;Autonomous Robots;2020-08-08
4. Online Disjoint Set Cover Without Prior Knowledge;LEIBNIZ INT PR INFOR;2019