Euclidean Bounded-Degree Spanning Tree Ratios

Let <i>t<sub>K</sub></i> be the worst-case (supremum) ratio of the weight of the minimum degree-<i>K</i> spanning tree to the weight of the minimum spanning tree, over all finite point sets in the Euclidean plane. It is known that <i>t<sub>2</sub>=2</i> and <i>t<sub>5</sub>=1</i>. In STOC'94, Khuller, Raghavachari, and Young established the following inequalities: <i>1.103<t<sub>3</sub>= 1.5</i> and <i>1.035<t<sub>4</sub>= 1.25</i>. We present the first improved upper bounds: <i>t<sub>3</sub> < 1.402</i> and <i>t<sub>4</sub> < 1.143</i>. As a result, we obtain better approximation algorithms for Euclidean minimum bounded-degree spanning trees.Let <i>t<sub>K</sub><sup>(d)</sup></i> be the analogous ratio in <i>d</i>-dimensional space. Khuller et al. showed that <i>t<sub>3</sub><sup>(d)</sup><1.667</i> for any <i>d</i>. We observe that <i>t<sub>3</sub><sup>(d)</sup><1.633</i>.

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

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

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

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

[5]  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..

[6]  Balaji Raghavachari,et al.  Algorithms for finding low degree structures , 1996 .

[7]  David Eppstein,et al.  Spanning Trees and Spanners , 2000, Handbook of Computational Geometry.

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

[9]  Thomas H. Cormen,et al.  Introduction to algorithms [2nd ed.] , 2001 .

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

[11]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[12]  Nicos Christofides.  Worst-Case Analysis of a New Heuristic for the Travelling Salesman Problem , 1976, Operations Research Forum.

[13]  Ding-Zhu Du,et al.  A proof of the Gilbert-Pollak conjecture on the Steiner ratio , 1992, Algorithmica.

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

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

[16]  D. Eppstein,et al.  Approximation algorithms for geometric problems , 1996 .

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

[18]  J. N. Lillington.  Some extremal properties of convex sets , 1975 .

[19]  Sándor P. Fekete,et al.  On Minimum Stars and Maximum Matchings , 2000, Discret. Comput. Geom..