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Encyclopedia of Algorithms pp 516–519Cite as

Degree-Bounded Trees

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  • First Online:

  • 119 Accesses

Years and Authors of Summarized Original Work

  • 1994; Fürer, Raghavachari

Problem Definition

The problem is to construct a spanning tree of small degree for a connected undirected graph \( { G=(V,E) } \). In the Steiner version of the problem, a set of distinguished vertices \( { D\subseteq V } \) is given along with the input graph G. A Steiner tree is a tree in G which spans at least the set D.

As finding a spanning or Steiner tree of the smallest possible degree \( { \Delta^* } \) is NP-hard, one is interested in approximating this minimization problem. For many such combinatorial optimization problems, the goal is to find an approximation in polynomial time (a constant or larger factor). For the spanning and Steiner tree problems, the iterative polynomial time approximation algorithms of Fürer and Raghavachari [8] (see also [14]) find much better solutions. The degree Δ of the solution tree is at most \( { \Delta^* +1 } \).

There are very few natural NP-hard optimization problems...

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Recommended Reading

  1. Chan TM (2004) Euclidean bounded-degree spanning tree ratios. Discret Comput Geom 32(2):177–194

    Article  MathSciNet  MATH  Google Scholar 

  2. Chaudhuri K, Rao S, Riesenfeld S, Talwar K (2006) A push-relabel algorithm for approximating degree bounded MSTs. In: Proceedings of the 33rd international colloquium on automata, languages and programming (ICALP 2006), Part I. LNCS, vol 4051. Springer, Berlin, pp 191–201

    Google Scholar 

  3. Chvátal V (1973) Tough graphs and Hamiltonian circuits. Discret Math 5:215–228

    Article  MathSciNet  MATH  Google Scholar 

  4. Fekete SP, Khuller S, Klemmstein M, Raghavachari B, Young N (1997) A network-flow technique for finding low-weightbounded-degree spanning trees. In: Proceedings of the 5th integer programming and combinatorial optimization conference (IPCO 1996), and J Algorithms 24(2):310–324

    Google Scholar 

  5. Fischer T (1993) Optimizing the degree of minimum weight spanning trees. Technical report TR93–1338. Computer Science Department, Cornell University

    Google Scholar 

  6. Fürer M, Raghavachari B (1990) An NC approximation algorithm for the minimum-degree spanning tree problem. In: Proceedings of the 28th annual Allerton conference on communication, control and computing, 1990, pp 174–281

    Google Scholar 

  7. Fürer M, Raghavachari B (1992) Approximating the minimum degree spanning tree to within one from the optimal degree. In: Proceedings of the third annual ACM-SIAM symposium on discrete algorithms (SODA 1992), pp 317–324

    Google Scholar 

  8. Fürer M, Raghavachari B (1994) Approximating the minimum-degree Steiner tree to within one of optimal. J Algorithms 17(3):409–423

    Article  MathSciNet  MATH  Google Scholar 

  9. Goemans MX (2006) Minimum bounded degree spanning trees. In: Proceedings of the 47th annual IEEE symposium on foundations of computer science (FOCS 2006), pp 273–282

    Google Scholar 

  10. Khuller S, Raghavachari B, Young N (1996) Low-degree spanning trees of small weight. SIAM J Comput 25(2):355–368

    Article  MathSciNet  MATH  Google Scholar 

  11. Klein PN, Krishnan R, Raghavachari B, Ravi R (2004) Approximation algorithms for finding low-degree subgraphs. Networks 44(3):203–215

    Article  MathSciNet  MATH  Google Scholar 

  12. Könemann J, Ravi R (2002) A matter of degree: improved approximation algorithms for degree-bounded minimum spanning trees. SIAM J Comput 31(6):1783–1793

    Article  MathSciNet  MATH  Google Scholar 

  13. Könemann J, Ravi R (2005) Primal-dual meets local search: approximating MSTs with nonuniform degree bounds. SIAM J Comput 34(3):763–773

    Article  MathSciNet  MATH  Google Scholar 

  14. Raghavachari B (1995) Algorithms for finding low degree structures. In: Hochbaum DS (ed) Approximation algorithms for NP-hard problems. PWS Publishing Company, Boston, pp 266–295

    Google Scholar 

  15. Ravi R, Singh M (2006) Delegate and conquer: an LP-based approximation algorithm for minimum degree MSTs. In: Proceedings of the 33rd international colloquiumon automata, languages and programming (ICALP 2006) Part I. LNCS, vol 4051. Springer, Berlin, pp 169–180

    Google Scholar 

  16. Singh M, Lau LC (2007) Approximating minimum bounded degree spanning trees to within one of optimal. In: Proceedings of the thirty-ninth annual ACM symposiumon theory of computing (STOC 2007), New York, pp 661–670

    Google Scholar 

  17. Win S (1989) On a connection between the existence of k-trees and the toughness of a graph. Graphs Comb 5(1):201–205

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, The Pennsylvania State University, University Park, PA, USA

    Martin Fürer

Authors
  1. Martin Fürer
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Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL, USA

    Ming-Yang Kao

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Fürer, M. (2016). Degree-Bounded Trees. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_104

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