TY - JOUR

T1 - Approximating Spanners and Directed Steiner Forest

AU - Chlamtáč, Eden

AU - Dinitz, Michael

AU - Kortsarz, Guy

AU - Laekhanukit, Bundit

N1 - Funding Information:
The preliminary version of this article [19] was published in Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’17). This work is supported by the National Science Foundation under Grant Nos. 1464239, 1535887 1218620, 1540547, and 1910565, by the Israel Science Foundation under Grant No. 1002/14 and 621/12, and Israeli Centers for Research Excellence under Grant No. 4/11. Authors’ addresses: E. Chlamtáč, Ben Gurion University, Department of Computer Science, P.O. Box 653, Be’er-Sheva, 84105, Israel; email: chlamtac@cs.bgu.ac.il; M. Dinitz, Johns Hopkins University, Department of Computer Science, 3400 N Charles Street, Baltimore, MD, 21218; email: mdinitz@cs.jhu.edu; G. Kortsarz, Rutgers University-Camden, Department of Computer Science, 227 Penn Street, Camden, NJ, 08102; email: guyk@camden.rutgers.edu; B. Laekhanukit, Shanghai University of Finance & Economics, School of Information Management & Engineering, 100 Wudong Road Yangpu District, Shanghai, 200433, China; email: bundit@sufe.edu.cn. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. © 2020 Association for Computing Machinery. 1549-6325/2020/06-ART33 $15.00 https://doi.org/10.1145/3381451
Publisher Copyright:
© 2020 ACM.

PY - 2020/6/1

Y1 - 2020/6/1

N2 - It was recently found that there are very close connections between the existence of additive spanners (subgraphs where all distances are preserved up to an additive stretch), distance preservers (subgraphs in which demand pairs have their distance preserved exactly), and pairwise spanners (subgraphs in which demand pairs have their distance preserved up to a multiplicative or additive stretch) [Abboud-Bodwin SODA'16 8 J.ACM'17, Bodwin-Williams SODA'16]. We study these problems from an optimization point of view, where rather than studying the existence of extremal instances, we are given an instance and are asked to find the sparsest possible spanner/preserver. We give an O(n3/5 + ϵ)-approximation for distance preservers and pairwise spanners (for arbitrary constant ϵ > 0). This is the first nontrivial upper bound for either problem, both of which are known to be as hard to approximate as Label Cover. We also prove Label Cover hardness for approximating additive spanners, even for the cases of additive 1 stretch (where one might expect a polylogarithmic approximation, since the related multiplicative 2-spanner problem admits an O(log n)-approximation) and additive polylogarithmic stretch (where the related multiplicative spanner problem has an O(1)-approximation). Interestingly, the techniques we use in our approximation algorithm extend beyond distance-based problem to pure connectivity network design problems. In particular, our techniques allow us to give an O(n3/5 + ϵ)-approximation for the Directed Steiner Forest problem (for arbitrary constant ϵ > 0) when all edges have uniform costs, improving the previous best O(n2/3 + ϵ)-approximation due to Berman et al. [ICALP'11] (which holds for general edge costs).

AB - It was recently found that there are very close connections between the existence of additive spanners (subgraphs where all distances are preserved up to an additive stretch), distance preservers (subgraphs in which demand pairs have their distance preserved exactly), and pairwise spanners (subgraphs in which demand pairs have their distance preserved up to a multiplicative or additive stretch) [Abboud-Bodwin SODA'16 8 J.ACM'17, Bodwin-Williams SODA'16]. We study these problems from an optimization point of view, where rather than studying the existence of extremal instances, we are given an instance and are asked to find the sparsest possible spanner/preserver. We give an O(n3/5 + ϵ)-approximation for distance preservers and pairwise spanners (for arbitrary constant ϵ > 0). This is the first nontrivial upper bound for either problem, both of which are known to be as hard to approximate as Label Cover. We also prove Label Cover hardness for approximating additive spanners, even for the cases of additive 1 stretch (where one might expect a polylogarithmic approximation, since the related multiplicative 2-spanner problem admits an O(log n)-approximation) and additive polylogarithmic stretch (where the related multiplicative spanner problem has an O(1)-approximation). Interestingly, the techniques we use in our approximation algorithm extend beyond distance-based problem to pure connectivity network design problems. In particular, our techniques allow us to give an O(n3/5 + ϵ)-approximation for the Directed Steiner Forest problem (for arbitrary constant ϵ > 0) when all edges have uniform costs, improving the previous best O(n2/3 + ϵ)-approximation due to Berman et al. [ICALP'11] (which holds for general edge costs).

KW - Approximation algorithms

KW - directed Steiner forest

KW - directed spanner

KW - hardness of approximation

KW - network design

UR - http://www.scopus.com/inward/record.url?scp=85088690469&partnerID=8YFLogxK

U2 - 10.1145/3381451

DO - 10.1145/3381451

M3 - Article

AN - SCOPUS:85088690469

SN - 1549-6325

VL - 16

JO - ACM Transactions on Algorithms

JF - ACM Transactions on Algorithms

IS - 3

M1 - 33

ER -