skip to main content
research-article
Public Access

Improved Approximation Algorithm for Steiner k-Forest with Nearly Uniform Weights

Published: 13 July 2017 Publication History

Abstract

In the Steiner k-Forest problem, we are given an edge weighted graph, a collection D of node pairs, and an integer k ⩽ |D|. The goal is to find a min-weight subgraph that connects at least k pairs. The best known ratio for this problem is min {O(√n), O(√k)} [Gupta et al. 2010]. In Gupta et al. [2010], it is also shown that ratio ρ for Steiner k-Forest implies ratio O(ρ · log 2n) for the related Dial-a-Ride problem. The only other algorithm known for Dial-a-Ride, besides the one resulting from Gupta et al. [2010], has ratio O(√n) [Charikar and Raghavachari 1998].
We obtain approximation ratio n0.448 for Steiner k-Forest and Dial-a-Ride with unit weights, breaking the O(√n) approximation barrier for this natural case. We also show that if the maximum edge-weight is O(nϵ), then one can achieve ratio O(n(1 + ϵ) · 0.448), which is less than √n if ϵ is small enough. The improvement for Dial-a-Ride is the first progress for this problem in 15 years. To prove our main result, we consider the following generalization of the Minimum k-Edge Subgraph (Mk-ES) problem, which we call Min-Cost ℓ-Edge-Profit Subgraph (MCℓ-EPS): Given a graph G = (V, E) with edge-profits p = {pe: eE} and node-costs c = {cv: vV}, and a lower profit bound ℓ, find a minimum node-cost subgraph of G of edge-profit at least ℓ. The Mk-ES problem is a special case of MCℓ-EPS with unit node costs and unit edge profits. The currently best known ratio for Mk-ES is n3-2√2 + ϵ [Chlamtac et al. 2012]. We extend this ratio to MCℓ-EPS for general node costs and profits bounded by a polynomial in n, which may be of independent interest.

References

[1]
A. Agrawal, P. Klein, and R. Ravi. 1995. When trees collide: An approximation algorithm for the generalized Steiner problem on networks. SIAM J. Comput. 24, 3, 440--456.
[2]
I. Althöfer, G. Das, D. P. Dobkin, D. Joseph, and J. Soares. 1993. On sparse spanners of weighted graphs. Discrete Comput. Geom. 9, 81--100.
[3]
B. Awerbuch. 1985. Complexity of network synchronization. J. ACM 32, 4, 804--823.
[4]
P. Berman and M. Karpinski. 2006. 8/7-approximation algorithm for (1, 2)-tsp. In Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’06), Miami, FL. 641--648.
[5]
A. Bhaskara, M. Charikar, E. Chlamtac, U. Feige, and A. Vijayaraghavan. 2010. Detecting high log-densities: an O(n1/4) approximation for densest k-subgraph. In Proceedings of the Symposium on Theory of Computing (STOC’10). 201--210.
[6]
J. Byrka, F. Grandoni, T. Rothvoß, and L. Sanità. 2013. Steiner tree approximation via iterative randomized rounding. J. ACM 60, 1, 6.
[7]
M. Charikar and B. Raghavachari. 1998. The finite capacity dial-a-ride problem. In Proceedings of the Annual Symposium on Foundations of Computer Science (FOCS’98). 458--467.
[8]
E. Chlamtac, M. Dinitz, and R. Krauthgamer. 2012. Everywhere-sparse spanners via dense subgraphs. In Proceedings of the Annual Symposium on Foundations of Computer Science (FOCS’12). 758--767.
[9]
J. Chuzhoy and J. Naor. 2006. Covering problems with hard capacities. SIAM J. Comput. 36, 2, 498--515.
[10]
U. Feige, G. Kortsarz, and D. Peleg. 2001. The dense k-subgraph problem. Algorithmica 29, 3, 410--421.
[11]
M. Feldman, G. Kortsarz, and Z. Nutov. 2012. Improved approximation algorithms for directed steiner forest. J. Comput. Syst. Sci. 78, 1, 279--292.
[12]
R. Gandhi, E. Halperin, S. Khuller, G. Kortsarz, and A. Srinivasan. 2006. An improved approximation algorithm for vertex cover with hard capacities. J. Comput. Syst. Sci. 72, 1, 16--33.
[13]
N. Garg. 2005. Saving an epsilon: A 2-approximation for the k-MST problem in graphs. In Proceedings of the Symposium on Theory of Computing (STOC’05). 396--402.
[14]
A. Gupta, M. T. Hajiaghayi, V. Nagarajan, and R. Ravi. 2010. Dial a ride from k-forest. ACM Trans. Algor. 6, 2.
[15]
M. T. Hajiaghayi and K. Jain. 2006. The prize-collecting generalized Steiner tree problem via a new approach of primal-dual schema. In Proceedings of the Symposium on Discrete Algorithms (SODA’06). 631--640.
[16]
G. Kortsarz and D. Peleg. 1993. On choosing a dense subgraph. In Proceedings of the Annual Symposium on Foundations of Computer Science (FOCS’93). 692--701.

Cited By

View all
  • (2020)Efficient and Simple Algorithms for Fault-Tolerant SpannersProceedings of the 39th Symposium on Principles of Distributed Computing10.1145/3382734.3405735(493-500)Online publication date: 31-Jul-2020
  • (2018)Leveraging content similarity among VMI files to allocate virtual machines in cloudFuture Generation Computer Systems10.1016/j.future.2017.09.05879:P2(528-542)Online publication date: 1-Feb-2018

Index Terms

  1. Improved Approximation Algorithm for Steiner k-Forest with Nearly Uniform Weights

      Recommendations

      Comments

      Information & Contributors

      Information

      Published In

      cover image ACM Transactions on Algorithms
      ACM Transactions on Algorithms  Volume 13, Issue 3
      July 2017
      390 pages
      ISSN:1549-6325
      EISSN:1549-6333
      DOI:10.1145/3058789
      Issue’s Table of Contents
      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 [email protected]

      Publisher

      Association for Computing Machinery

      New York, NY, United States

      Publication History

      Published: 13 July 2017
      Accepted: 01 March 2017
      Revised: 01 January 2017
      Received: 01 May 2015
      Published in TALG Volume 13, Issue 3

      Permissions

      Request permissions for this article.

      Check for updates

      Author Tags

      1. Dial-a-Ride
      2. Minimum k-Edge Subgraph

      Qualifiers

      • Research-article
      • Research
      • Refereed

      Funding Sources

      Contributors

      Other Metrics

      Bibliometrics & Citations

      Bibliometrics

      Article Metrics

      • Downloads (Last 12 months)39
      • Downloads (Last 6 weeks)8
      Reflects downloads up to 17 Jan 2025

      Other Metrics

      Citations

      Cited By

      View all
      • (2020)Efficient and Simple Algorithms for Fault-Tolerant SpannersProceedings of the 39th Symposium on Principles of Distributed Computing10.1145/3382734.3405735(493-500)Online publication date: 31-Jul-2020
      • (2018)Leveraging content similarity among VMI files to allocate virtual machines in cloudFuture Generation Computer Systems10.1016/j.future.2017.09.05879:P2(528-542)Online publication date: 1-Feb-2018

      View Options

      View options

      PDF

      View or Download as a PDF file.

      PDF

      eReader

      View online with eReader.

      eReader

      Login options

      Full Access

      Media

      Figures

      Other

      Tables

      Share

      Share

      Share this Publication link

      Share on social media