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 the problem is open for @d=4. The problem is polynomial-time solvable when @d=5. By presenting an improved approximation analysis for Chan's degree-4 MST algorithm [T. Chan, Euclidean bounded-degree spanning tree ratios, Discrete & Computational Geometry 32 (2004) 177-194], we show that, for any arbitrary collection of points in the Euclidean plane, there always exists a degree-4 spanning tree of weight at most (2+2)/3<1.1381 times the weight of an MST.

[1]  Timothy M. Chan.  Euclidean Bounded-Degree Spanning Tree Ratios , 2003, SCG '03.

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

[3]  Sanjeev Arora,et al.  Approximation schemes for NP-hard geometric optimization problems: a survey , 2003, Math. Program..

[4]  Satish Rao,et al.  What Would Edmonds Do? Augmenting Paths and Witnesses for Degree-Bounded MSTs , 2005, APPROX-RANDOM.

[5]  Sanjeev Arora,et al.  Polynomial time approximation schemes for Euclidean TSP and other geometric problems , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[6]  Gautam Das,et al.  Constructing Degree-3 Spanners with Other Sparseness Properties , 1996, Int. J. Found. Comput. Sci..

[7]  Christos H. Papadimitriou,et al.  On Two Geometric Problems Related to the Traveling Salesman Problem , 1984, J. Algorithms.

[8]  Michel X. Goemans,et al.  Minimum Bounded Degree Spanning Trees , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[9]  Clyde L. Monma,et al.  Transitions in geometric minimum spanning trees , 1991, SCG '91.

[10]  R. Ravi,et al.  Primal-Dual Meets Local Search: Approximating MSTs With Nonuniform Degree Bounds , 2005, SIAM J. Comput..

[11]  Joseph S. B. Mitchell,et al.  Guillotine Subdivisions Approximate Polygonal Subdivisions: A Simple Polynomial-Time Approximation Scheme for Geometric TSP, k-MST, and Related Problems , 1999, SIAM J. Comput..

[12]  R. Ravi,et al.  A matter of degree: improved approximation algorithms for degree-bounded minimum spanning trees , 2000, STOC '00.

[13]  Sanjeev Arora,et al.  Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems , 1998, JACM.

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

[15]  Jeffrey S. Salowe.  Euclidean Spanner Graphs with Degree Four , 1994, Discret. Appl. Math..

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

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

[18]  Jayme Luiz Szwarcfiter,et al.  Hamilton Paths in Grid Graphs , 1982, SIAM J. Comput..

[19]  Sanjeev Arora,et al.  Approximation Schemes for Degree-Restricted MST and Red–Blue Separation Problems , 2004, Algorithmica.

[20]  Jeffrey S. Salowe,et al.  Low-degree minimum spanning trees , 1995, Discret. Comput. Geom..

[21]  Kevin L. Chang,et al.  Approximation Schemes for Degree-Restricted MST and Red-Blue Separation Problem , 2003, ICALP.

[22]  Samir Khuller,et al.  A Network-Flow Technique for Finding Low-Weight Bounded-Degree Spanning Trees , 1996, J. Algorithms.