research-article

A Unified PTAS for Prize Collecting TSP and Steiner Tree Problem in Doubling Metrics

Authors:

T.-H. Hubert Chan,

Shaofeng H.-C. JiangAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 16, Issue 2

Article No.: 24, Pages 1 - 23

https://doi.org/10.1145/3378571

Published: 09 March 2020 Publication History

Abstract

We present a unified (randomized) polynomial-time approximation scheme (PTAS) for the prize collecting traveling salesman problem (PCTSP) and the prize collecting Steiner tree problem (PCSTP) in doubling metrics. Given a metric space and a penalty function on a subset of points known as terminals, a solution is a subgraph on points in the metric space whose cost is the weight of its edges plus the penalty due to terminals not covered by the subgraph. Under our unified framework, the solution subgraph needs to be Eulerian for PCTSP, while it needs to be a tree for PCSTP. Before our work, even a QPTAS for the problems in doubling metrics is not known.

Our unified PTAS is based on the previous dynamic programming frameworks proposed in Talwar (STOC 2004) and Bartal, Gottlieb, Krauthgamer (STOC 2012). However, since it is unknown which part of the optimal cost is due to edge lengths and which part is due to penalties of uncovered terminals, we need to develop new techniques to apply previous divide-and-conquer strategies and sparse instance decompositions.

References

[1]

Ittai Abraham, Yair Bartal, and Ofer Neiman. 2006. Advances in metric embedding theory. In Proceedings of the STOC. ACM, 271--286.

Digital Library

[2]

Aaron Archer, Mohammad Hossein Bateni, Mohammad Taghi Hajiaghayi, and Howard J. Karloff. 2011. Improved approximation algorithms for prize-collecting Steiner tree and TSP. SIAM J. Comput. 40, 2 (2011), 309--332.

Digital Library

[3]

Sanjeev Arora. 1998. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM 45, 5 (1998), 753--782.

Digital Library

[4]

P. Assouad. 1983. Plongements lipschitziens dans . Bull. Soc. Math. France 111, 4 (1983), 429--448. BSMFAA

[5]

Egon Balas. 1989. The prize collecting traveling salesman problem. Networks 19, 6 (1989), 621--636.

[6]

Yair Bartal, Lee-Ad Gottlieb, and Robert Krauthgamer. 2016. The traveling salesman problem: Low-dimensionality implies a polynomial time approximation scheme. SIAM J. Comput. 45, 4 (2016), 1563--1581.

[7]

Mohammad Hossein Bateni, Chandra Chekuri, Alina Ene, Mohammad Taghi Hajiaghayi, Nitish Korula, and Dániel Marx. 2011. Prize-collecting Steiner problems on planar graphs. In Proceedings of the SODA. SIAM, 1028--1049.

[8]

Mohammad Hossein Bateni and MohammadTaghi Hajiaghayi. 2012. Euclidean prize-collecting Steiner forest. Algorithmica 62, 3--4 (2012), 906--929.

Digital Library

[9]

Daniel Bienstock, Michel X. Goemans, David Simchi-Levi, and David Williamson. 1993. A note on the prize collecting traveling salesman problem. Math. Prog. 59, 1 (1993), 413--420.

[10]

Glencora Borradaile, Philip N. Klein, and Claire Mathieu. 2015. A polynomial-time approximation scheme for Euclidean Steiner forest. ACM Trans. Algor. 11, 3 (2015), 19:1--19:20.

[11]

T.-H. Hubert Chan, Shuguang Hu, and Shaofeng H.-C. Jiang. 2018. A PTAS for the Steiner forest problem in doubling metrics. SIAM J. Comput. 47, 4 (2018), 1705--1734.

[12]

T.-H. Hubert Chan and Shaofeng H.-C. Jiang. 2018. Reducing curse of dimensionality: Improved PTAS for TSP (with neighborhoods) in doubling metrics. ACM Trans. Algor. 14, 1 (2018), 9:1--9:18.

[13]

T.-H. Hubert Chan, Mingfei Li, Li Ning, and Shay Solomon. 2015. New doubling spanners: Better and simpler. SIAM J. Comput. 44, 1 (2015), 37--53.

Digital Library

[14]

Kenneth L. Clarkson. 1999. Nearest neighbor queries in metric spaces. Disc. Computat. Geom. 22, 1 (1999), 63--93.

[15]

Erik D. Demaine, Mohammad Taghi Hajiaghayi, and Ken-ichi Kawarabayashi. 2011. Contraction decomposition in h-minor-free graphs and algorithmic applications. In Proceedings of the STOC. ACM, 441--450.

Digital Library

[16]

M. M. Deza and M. Laurent. 1997. Geometry of Cuts and Metrics. Algorithms and Combinatorics, Vol. 15. Springer-Verlag, Berlin. xii+587 pages.

[17]

Michel X. Goemans. 2009. Combining approximation algorithms for the prize-collecting TSP. CoRR abs/0910.0553 (2009).

[18]

Michel X. Goemans and David P. Williamson. 1995. A general approximation technique for constrained forest problems. SIAM J. Comput. 24, 2 (1995), 296--317.

Digital Library

[19]

Anupam Gupta, Robert Krauthgamer, and James R. Lee. 2003. Bounded geometries, fractals, and low-distortion embeddings. In Proceedings of the FOCS. IEEE Computer Society, 534--543.

Digital Library

[20]

J. Matoušek. 2002. Lectures on Discrete Geometry. Graduate Texts in Mathematics, Vol. 212. Springer-Verlag, New York. xvi+481 pages.

[21]

Kunal Talwar. 2004. Bypassing the embedding: Algorithms for low dimensional metrics. In Proceedings of the STOC. ACM, 281--290.

Digital Library

Cited By

Blauth JKlein NNägele M(2024)A Better-Than-1.6-Approximation for Prize-Collecting TSPInteger Programming and Combinatorial Optimization10.1007/978-3-031-59835-7_3(28-42)Online publication date: 3-Jul-2024
https://dl.acm.org/doi/10.1007/978-3-031-59835-7_3
Blauth JNägele MSaha BServedio R(2023)An Improved Approximation Guarantee for Prize-Collecting TSPProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585159(1848-1861)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585159

Index Terms

A Unified PTAS for Prize Collecting TSP and Steiner Tree Problem in Doubling Metrics
1. Theory of computation
  1. Design and analysis of algorithms
    1. Approximation algorithms analysis
      1. Routing and network design problems

Recommendations

Improved Approximation Algorithms for PRIZE-COLLECTING STEINER TREE and TSP
FOCS '09: Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science

We study the prize-collecting versions of the Steiner tree, traveling salesman, and stroll (a.k.a. Path-TSP) problems (PCST, PCTSP, and PCS, respectively): given a graph (V, E) with costs on each edge and a penalty (a.k.a. prize) on each node, the goal ...
Prize-collecting Steiner problems on planar graphs
SODA '11: Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete algorithms

In this paper, we reduce Prize-Collecting Steiner TSP (PCTSP), Prize-Collecting Stroll (PCS), Prize-Collecting Steiner Tree (PCST), Prize-Collecting Steiner Forest (PCSF), and more generally Submodular Prize-Collecting Steiner Forest (SPCSF), on planar ...
Variations of the prize-collecting Steiner tree problem

The prize-collecting Steiner tree problem is well known to be NP-hard. We consider seven variations of this problem generalizing several well-studied bottleneck and minsum problems with feasible solutions as trees of a graph. Four of these problems are ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 16, Issue 2

April 2020

372 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/3386689

Editor:
Aravind Srinivasan
University of Maryland, USA

Issue’s Table of Contents

Copyright © 2020 ACM.

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 the author(s) 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: 09 March 2020

Revised: 01 August 2019

Accepted: 01 January 2019

Received: 01 October 2018

Published in TALG Volume 16, Issue 2

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed

Funding Sources

Hong Kong RGC

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

2
Total Citations
View Citations
207
Total Downloads

Downloads (Last 12 months)17
Downloads (Last 6 weeks)2

Reflects downloads up to 26 Jul 2024

Other Metrics

View Author Metrics

Citations

Cited By

Blauth JKlein NNägele M(2024)A Better-Than-1.6-Approximation for Prize-Collecting TSPInteger Programming and Combinatorial Optimization10.1007/978-3-031-59835-7_3(28-42)Online publication date: 3-Jul-2024
https://dl.acm.org/doi/10.1007/978-3-031-59835-7_3
Blauth JNägele MSaha BServedio R(2023)An Improved Approximation Guarantee for Prize-Collecting TSPProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585159(1848-1861)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585159

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

HTML Format

View this article in HTML Format.

Media

Figures

Other

Tables

View Issue’s Table of Contents