A matter of degree: improved approximation algorithms for degree-bounded minimum spanning trees

In this paper, we present a new bicriteria approximation algorithm for the degree- bounded minimum spanning tree problem. In this problem, we are given an undirected graph, a nonnegative cost function on the edges, and a positive integer B ∗ , and the goal is to find a minimum- cost spanning tree T with maximum degree at most B ∗ .I n ann-node graph, our algorithm finds a spanning tree with maximum degree O(B∗ + log n) and cost O( opt B∗ ), where opt B∗ is the minimum cost of any spanning tree whose maximum degree is at most B ∗ . Our algorithm uses ideas from Lagrangean duality. We show how a set of optimum Lagrangean multipliers yields bounds on both the degree and the cost of the computed solution.

[1]  Philip Wolfe,et al.  Validation of subgradient optimization , 1974, Math. Program..

[2]  R. Ravi,et al.  Many birds with one stone: multi-objective approximation algorithms , 1993, STOC '93.

[3]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .

[4]  Dorit S. Hochbaum,et al.  Approximation Algorithms for NP-Hard Problems , 1996 .

[5]  R. Ravi,et al.  A nearly best-possible approximation algorithm for node-weighted Steiner trees , 1993, IPCO.

[6]  R. Ravi,et al.  Bicriteria Network Design Problems , 1998, J. Algorithms.

[7]  Andrew S. Tanenbaum,et al.  Computer networks (3rd ed.) , 1996 .

[8]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[9]  S. Chopra.  On the spanning tree polyhedron , 1989 .

[10]  Ted Fischer,et al.  Optimizing the Degree of Minimum Weight Spanning Trees , 1993 .

[11]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[12]  Andrew S. Tanenbaum,et al.  Computer Networks , 1981 .

[13]  Martin Fürer,et al.  Approximating the Minimum-Degree Steiner Tree to within One of Optimal , 1994, J. Algorithms.

[14]  Deborah Estrin,et al.  An architecture for wide-area multicast routing , 1994, SIGCOMM 1994.

[15]  Richard M. Karp,et al.  The traveling-salesman problem and minimum spanning trees: Part II , 1971, Math. Program..

[16]  Yoshikuni Onozato,et al.  A copy network with shared buffers for large-scale multicast ATM switching , 1993, TNET.

[17]  J. Davenport.  Editor , 1960 .

[18]  Anujan Varma,et al.  Degree-constrained multicasting in point-to-point networks , 1995, Proceedings of INFOCOM'95.

[19]  Deborah Estrin,et al.  The PIM architecture for wide-area multicast routing , 1996, TNET.

[20]  Dorit S. Hochba,et al.  Approximation Algorithms for NP-Hard Problems , 1997, SIGA.

[21]  Samir Khuller,et al.  Low-Degree Spanning Trees of Small Weight , 1996, SIAM J. Comput..

[22]  M. Gerla,et al.  Multicasting protocols for high-speed, wormhole-routing local area networks , 1996, SIGCOMM '96.

[23]  Steven McCanne,et al.  A reliable multicast framework for light-weight sessions and application level framing , 1995, SIGCOMM '95.