Algorithmica

, Volume 55, Issue 1, pp 157–189 | Cite as

What Would Edmonds Do? Augmenting Paths and Witnesses for Degree-Bounded MSTs

  • Kamalika Chaudhuri
  • Satish Rao
  • Samantha Riesenfeld
  • Kunal Talwar
Article

Abstract

Given a graph and degree upper bounds on vertices, the BDMST problem requires us to find a minimum cost spanning tree respecting the given degree bounds. This problem generalizes the Travelling Salesman Path Problem (TSPP), even in unweighted graphs, and so we expect that it is necessary to relax the degree constraints to get efficient algorithms. Könemann and Ravi (Proceedings of the Thirty Second Annual ACM Symposium on Theory of Computing, pp. 537–546, 2000; Proceedings of the Thirty-Fifth ACM Symposium on Theory of Computing, pp. 389–395, 2003) give bicriteria approximation algorithms for the problem using local search techniques of Fischer (Technical Report 14853, Cornell University, 1993). Their algorithms find solutions which make a tradeoff of the approximation factor for the cost of the resulting tree against the factor by which degree constraints are violated. In particular, they give an algorithm which, for a graph with a spanning tree of cost C and degree B, and for parameters b,w>1, produces a tree whose cost is at most wC and whose degree is at most \(\frac{w}{w-1}bB+\log_{b}n.\)

A primary contribution of Könemann and Ravi is to use a Lagrangean relaxation to formally relate the BDMST problem to what we call the MDMST problem, which is the problem of finding an MST of minimum degree in a graph. In their solution to the MDMST problem, they make central use of a local-search approximation algorithm of Fischer.

In this paper, we give the first approximation algorithms for the BDMST problem—both our algorithms find trees of optimal cost. We achieve this improvement using a primal-dual cost bounding methodology from Edmonds’ weighted matching algorithms which was not previously used in this context. In order to follow Edmonds’ approach, we develop algorithms for a variant of the MDMST problem in which there are degree lower bound requirements. This variant may be of independent interest; in particular, our results extend to a generalized version of the BDMST problem in which both upper and lower degree bounds are given.

First we give a polynomial-time algorithm that finds a tree of optimal cost and with maximum degree at most \(\frac{b}{2-b}B+O(\log_{b}n)\) for any b∈(1,2). We also give a quasi-polynomial-time approximation algorithm which produces a tree of optimal cost C and maximum degree at most B+O(log n/log log n). That is, the error is additive as well as restricted just to the degree. This further improvement in degree is obtained by using augmenting-path techniques that search over a larger solution space than Fischer’s local-search algorithm.

Keywords

Approximation algorithms Combinatorial optimization Minimum spanning trees Matching 

References

  1. 1.
    Chan, T.M.: Euclidean bounded-degree spanning tree ratios. In: Proceedings of the Nineteenth Annual Symposium on Computational Geometry, pp. 11–19. ACM Press, New York (2003) CrossRefGoogle Scholar
  2. 2.
    Chaudhuri, K., Rao, S., Riesenfeld, S., Talwar, K.: What would Edmonds do? Augmenting paths and witnesses for degree-bounded msts. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX-RANDOM. Lecture Notes in Computer Science, vol. 3624, pp. 26–39. Springer, New York (2005) Google Scholar
  3. 3.
    Chaudhuri, K., Rao, S., Riesenfeld, S., Talwar, K.: A push-relabel algorithm for approximating degree bounded MSTs. In: ICALP (1). Lecture Notes in Computer Science, vol. 4051, pp. 191–201. Springer, New York (2006) Google Scholar
  4. 4.
    Edmonds, J.: Maximum matching and a polyhedron with 0–1 vertices. J. Res. Nat. Bur. Stand. 69B, 125–130 (1965) MathSciNetGoogle Scholar
  5. 5.
    Edmonds, J.: Paths, trees, and flowers. Can. J. Math. 17, 449–467 (1965) MATHMathSciNetGoogle Scholar
  6. 6.
    Ellingham, M., Zha, X.: Toughness, trees and walks. J. Graph Theory 33(3), 125–137 (2000) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Even, S., Tarjan, R.E.: Network flow and testing graph connectivity. SIAM J. Comput. 4(4), 507–518 (1975) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Fischer, T.: Optimizing the degree of minimum weight spanning trees. Technical Report 14853, Dept. of Computer Science, Cornell University, Ithaca, NY (1993) Google Scholar
  9. 9.
    Fürer, M., Raghavachari, B.: Approximating the minimum-degree Steiner tree to within one of optimal. J. Algorithms 17(3), 409–423 (1994) CrossRefMathSciNetGoogle Scholar
  10. 10.
    Gavish, B.: Topological design of centralized computer networks–formulations and algorithms. Networks 12, 355–377 (1982) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Goemans, M.X.: Minimum bounded degree spanning trees. In: FOCS, pp. 273–282. IEEE Computer Society, New York (2006) Google Scholar
  12. 12.
    Hoogeveen, J.A.: Analysis of Christofides’ heuristic: some paths are more difficult than cycles. Oper. Res. Lett. 10, 291–295 (1991) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hopcroft, J., Karp, R.: An n 5/2 algorithm for maximum matching in bipartite graphs. SIAM J. Comput. 2, 225–231 (1973) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Jothi, R., Raghavachari, B.: Degree-bounded minimum spanning trees. In: Proc. 16th Canadian Conf. on Computational Geometry (CCCG), 2004 Google Scholar
  15. 15.
    Khuller, S., Raghavachari, B., Young, N.: Low-degree spanning trees of small weight. SIAM J. Comput. 25(2), 355–368 (1996) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Könemann, J., Ravi, R.: A matter of degree: improved approximation algorithms for degree-bounded minimum spanning trees. In: Proceedings of the Thirty Second Annual ACM Symposium on Theory of Computing: Portland, Oregon, May 21–23, 2000, pp. 537–546. ACM Press, New York (2000) CrossRefGoogle Scholar
  17. 17.
    Könemann, J., Ravi, R.: Primal-dual meets local search: approximating MST’s with nonuniform degree bounds. In: Proceedings of the Thirty-Fifth ACM Symposium on Theory of Computing, San Diego, CA, USA, June 9–11, 2003, pp. 389–395. ACM Press, New York (2003) CrossRefGoogle Scholar
  18. 18.
    Krishnan, R., Raghavachari, B.: The directed minimum degree spanning tree problem. In: FSTTCS, pp. 232–243 (2001) Google Scholar
  19. 19.
    Papadimitriou, C.H., Vazirani, U.: On two geometric problems related to the traveling salesman problem. J. Algorithms 5, 231–246 (1984) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Ravi, R., Singh, M.: Delegate and conquer: an LP-based approximation algorithm for minimum degree MSTs. In: ICALP (1). Lecture Notes in Computer Science, vol. 4051, pp. 169–180. Springer, New York (2006) Google Scholar
  21. 21.
    Ravi, R., Marathe, M.V., Ravi, S.S., Rosenkrantz, D.J., Hunt, H.B. III: Many birds with one stone: multi-objective approximation algorithms. In: Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, pp. 438–447. ACM Press, New York (1993) CrossRefGoogle Scholar
  22. 22.
    Ravi, R., Marathe, M.V., Ravi, S.S., Rosenkrantz, D.J., Hunt, H.B. III: Approximation algorithms for degree-constrained minimum-cost network-design problems. Algorithmica 31 (2001) Google Scholar
  23. 23.
    Singh, M., Lau, L.C.: Approximating minimum bounded degree spanning trees to within one of optimal. In: STOC ’07: Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, pp. 661–670. ACM Press, New York (2007) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Kamalika Chaudhuri
    • 1
  • Satish Rao
    • 1
  • Samantha Riesenfeld
    • 1
  • Kunal Talwar
    • 2
  1. 1.University of CaliforniaBerkeleyUSA
  2. 2.Microsoft ResearchMountain ViewUSA

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