research-article

Approximating Minimum Bounded Degree Spanning Trees to within One of Optimal

Authors:

Lap Chi LauAuthors Info & Claims

Journal of the ACM (JACM), Volume 62, Issue 1

Article No.: 1, Pages 1 - 19

https://doi.org/10.1145/2629366

Published: 02 March 2015 Publication History

Abstract

In the Minimum Bounded Degree Spanning Tree problem, we are given an undirected graph G = (V, E) with a degree upper bound B_v on each vertex v ∈ V, and the task is to find a spanning tree of minimum cost that satisfies all the degree bounds. Let OPT be the cost of an optimal solution to this problem. In this article we present a polynomial-time algorithm which returns a spanning tree T of cost at most OPT and d_T(v) ≤ B_v + 1 for all v, where d_T(v) denotes the degree of v in T. This generalizes a result of Fürer and Raghavachari [1994] to weighted graphs, and settles a conjecture of Goemans [2006] affirmatively. The algorithm generalizes when each vertex v has a degree lower bound A_v and a degree upper bound B_v, and returns a spanning tree with cost at most OPT and A_v - 1 ≤ d_T(v) ≤ B_v + 1 for all v ∈ V. This is essentially the best possible. The main technique used is an extension of the iterative rounding method introduced by Jain [2001] for the design of approximation algorithms.

References

[1]

N. Bansal, R. Khandekar, and V. Nagarajan. 2008. Additive guarantees for degree bounded directed network design. In Proceedings of STOC. 769--778.

Digital Library

[2]

K. Chaudhuri, S. Rao, S. Riesenfeld, and K. Talwar. 2009a. A push-relabel approximation algorithm for approximating the minimum-degree MST problem and its generalization to matroids. Theoret. Comput. Sci. 410, 44, 4489--4503.

Digital Library

[3]

K. Chaudhuri, S. Rao, S. Riesenfeld, and K. Talwar. 2009b. What would Edmonds Do&quest; Augmenting paths and witnesses for degree-bounded MSTs. Algorithmica 55, 1, 157--189.

Digital Library

[4]

J. Cheriyan, S. Vempala, and A. Vetta. 2006. Network design via iterative rounding of setpair relaxations. Combinatorica 26, 3, 255--275.

Digital Library

[5]

G. Cornuéjols, J. Fonlupt, and D. Naddef. 1985. The traveling salesman problem on a graph and some related integer polyhedra. Math. Program. 33, 1--27. 10.1007/BF01582008.

Digital Library

[6]

W. H. Cunningham. 1984. Testing membership in matroid polyhedra. J. Combinat. Theory, Seri. B 36, 2, 161--188.

[7]

J. Edmonds. 1971. Matroids and the greedy algorithm. Math. Program. 1, 127--136. 10.1007/BF01584082.

Digital Library

[8]

L. Fleischer, K. Jain, and D. P. Williamson. 2006. Iterative rounding 2-approximation algorithms for minimum-cost vertex connectivity problems. J. Comput. Syst. Sci. 72, 5, 838--867.

Digital Library

[9]

M. Fürer and B. Raghavachari. 1994. Approximating the minimum-degree Steiner tree to within one of optimal. J. Algor. 17, 3, 409--423.

Digital Library

[10]

M. R. Garey and D. S. Johnson. 1979. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York.

Digital Library

[11]

M. X. Goemans. 2006. Minimum bounded degree spanning trees. In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science. 273--282.

Digital Library

[12]

M. Grötschel, L. Lovász, and A. Schrijver. 1981. The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 2, 169--197.

[13]

K. Jain. 2001. A Factor 2 approximation algorithm for the generalized steiner network problem. Combinatorica 21, 1, 39--60.

[14]

J. Könemann and R. Ravi. 2002. A matter of degree: Improved approximation algorithms for degree-bounded minimum spanning trees. SIAM J. Comput. 31, 6, 1783--1793.

Digital Library

[15]

J. Könemann and R. Ravi. 2005. Primal-dual meets local search: Approximating MSTs with nonuniform degree bounds. SIAM J. Comput. 34, 3, 763--773.

Digital Library

[16]

L. C. Lau, J. Naor, M. R. Salavatipour, and M. Singh. 2009. Survivable network design with degree or order constraints. SIAM J. Comput. 39, 3, 1062--1087.

Digital Library

[17]

L. C. Lau, R. Ravi, and M. Singh. 2011. Iterative Methods in Combinatorial Optimization. Cambridge Tects in Applied Mathematics, Cambridge University Press.

Digital Library

[18]

L. C. Lau and M. Singh. 2008. Additive approximation for bounded degree survivable network design. In Proceedings of STOC. 759--768.

Digital Library

[19]

R. Ravi, M. V. Marathe, S. S. Ravi, D. J. Rosenkrantz, and H. B. Hunt. 1993. Many birds with one stone: Multi-objective approximation algorithms. In Proceedings of the 25th Annual ACM Symposium on Theory of Computing. 438--447.

Digital Library

[20]

R. Ravi and M. Singh. 2006. Delegate and conquer: An LP-based approximation algorithm for minimum degree MSTs. In Proceedings of the 33rd International Colloquium on Automata, Languages and Programming. 169--180.

Digital Library

[21]

A. Schrijver. 2003. Combinatorial Optimization - Polyhedra and Efficiency. Springer.

[22]

M. Singh. 2008. Iterative Methods in Combinatorial Optimization. Ph.D. thesis Carnegie Mellon University.

[23]

V. V. Vazirani. 2004. Approximation Algorithms. Springer.

Digital Library

[24]

S. Win. 1989. On a connection between the existence OFK-trees and the toughness of a graph. Graphs and Combinatorics 5, 1, 201--205.

Cited By

Hershkowitz DKlein NZenklusen RMohar BShinkar IO'Donnell R(2024)Ghost Value Augmentation for k-Edge-ConnectivityProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649715(1853-1864)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649715
Huang ZLiao XNaik PLu X(2024)On approximating a new generalization of traveling salesman problemHeliyon10.1016/j.heliyon.2024.e3129710:10(e31297)Online publication date: May-2024
https://doi.org/10.1016/j.heliyon.2024.e31297
Shi YHan CGuo T(2024)NeuroPrim: An attention-based model for solving NP-hard spanning tree problemsScience China Mathematics10.1007/s11425-022-2175-567:6(1359-1376)Online publication date: 22-Feb-2024
https://doi.org/10.1007/s11425-022-2175-5
Show More Cited By

Index Terms

Approximating Minimum Bounded Degree Spanning Trees to within One of Optimal
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Network flows
      2. Trees
2. Theory of computation
  1. Design and analysis of algorithms
    1. Graph algorithms analysis
      1. Network flows
  2. Randomness, geometry and discrete structures

Recommendations

Approximating minimum bounded degree spanning trees to within one of optimal
STOC '07: Proceedings of the thirty-ninth annual ACM symposium on Theory of computing

In the Minimum Bounded Degree Spanning Tree problem, we aregiven an undirected graph with a degree upper bound B_v on eachvertex v, and the task is to find a spanning tree of minimumcost which satisfies all the degree bounds. Let OPT be the costof an ...
Low-Degree Spanning Trees of Small Weight

Given $n$ points in the plane, the degree-$K$ spanning-tree problem asks for a spanning tree of minimum weight in which the degree of each vertex is at most $K$. This paper addresses the problem of computing low-weight degree-$K$ spanning trees for $K>2$...
Degree-bounded minimum spanning trees

Given n points in the Euclidean plane, the degree-@d minimum spanning tree (MST) problem asks for a spanning tree of minimum weight in which the degree of each vertex is at most @d. The problem is NP-hard for 2@__ __@d@__ __3, while the NP-hardness of ...

Comments

Information & Contributors

Information

Published In

cover image Journal of the ACM

Journal of the ACM Volume 62, Issue 1

February 2015

264 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/2742144

Editor:
Victor Vianu
University of California, San Diego

Issue’s Table of Contents

Copyright © 2015 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 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: 02 March 2015

Accepted: 01 April 2014

Revised: 01 February 2014

Received: 01 April 2013

Published in JACM Volume 62, Issue 1

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

32
Total Citations
View Citations
955
Total Downloads

Downloads (Last 12 months)77
Downloads (Last 6 weeks)3

Reflects downloads up to 23 Dec 2024

Other Metrics

View Author Metrics

Citations

Cited By

Hershkowitz DKlein NZenklusen RMohar BShinkar IO'Donnell R(2024)Ghost Value Augmentation for k-Edge-ConnectivityProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649715(1853-1864)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649715
Huang ZLiao XNaik PLu X(2024)On approximating a new generalization of traveling salesman problemHeliyon10.1016/j.heliyon.2024.e3129710:10(e31297)Online publication date: May-2024
https://doi.org/10.1016/j.heliyon.2024.e31297
Shi YHan CGuo T(2024)NeuroPrim: An attention-based model for solving NP-hard spanning tree problemsScience China Mathematics10.1007/s11425-022-2175-567:6(1359-1376)Online publication date: 22-Feb-2024
https://doi.org/10.1007/s11425-022-2175-5
Klein NOlver N(2023)Thin Trees for Laminar Families2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00011(50-59)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00011
Bousquet NIto TKobayashi YMizuta HOuvrard PSuzuki AWasa K(2023)Reconfiguration of Spanning Trees with Degree Constraints or Diameter ConstraintsAlgorithmica10.1007/s00453-023-01117-z85:9(2779-2816)Online publication date: 10-Apr-2023
https://doi.org/10.1007/s00453-023-01117-z
Marbel RYozevitch RGrinshpoun TBen-Moshe B(2022)Dynamic Network Formation for FSO Satellite CommunicationApplied Sciences10.3390/app1202073812:2(738)Online publication date: 12-Jan-2022
https://doi.org/10.3390/app12020738
Faenza YKavitha T(2022)Quasi-Popular Matchings, Optimality, and Extended FormulationsMathematics of Operations Research10.1287/moor.2021.113947:1(427-457)Online publication date: 1-Feb-2022
https://dl.acm.org/doi/10.1287/moor.2021.1139
Bérczi KMnich MVincze R(2022)A 3/2-Approximation for the Metric Many-Visits Path TSPSIAM Journal on Discrete Mathematics10.1137/22M148341436:4(2995-3030)Online publication date: 1-Jan-2022
https://dl.acm.org/doi/10.1137/22M1483414
Lau LZhou H(2022)A Spectral Approach to Network DesignSIAM Journal on Computing10.1137/20M133076251:4(1018-1064)Online publication date: 1-Jan-2022
https://dl.acm.org/doi/10.1137/20M1330762
Laddha ASingh MVempala S(2022)Socially fair network design via iterative roundingOperations Research Letters10.1016/j.orl.2022.07.01150:5(536-540)Online publication date: Sep-2022
https://doi.org/10.1016/j.orl.2022.07.011
Show More Cited By

View Options

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.

Media

Figures

Other

Tables

View Issue’s Table of Contents