The algebraic degree of geometric optimization problems
Discrete & Computational Geometry
Algebraic optimization: the Fermat-Weber location problem
Mathematical Programming: Series A and B
The maximum number of second smallest distances in finite planar sets
Discrete & Computational Geometry
Fast approximations for sums of distances, clustering and the Fermat--Weber problem
Computational Geometry: Theory and Applications
Distinct Triangle Areas in a Planar Point Set
IPCO '07 Proceedings of the 12th international conference on Integer Programming and Combinatorial Optimization
On stars and Steiner stars: II
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
| Hi-index | 0.00 |
For a set of n points in the plane, a star connects one of the points (the center) to the other n - 1 points by straight line edges, while a Steiner star connects an arbitrary point in the plane to all n input points. The center of the minimum Steiner star, the Weber center, minimizes the sum of distances from the n points. Fekete and Meijer showed that the minimum star is at most √2 times longer than the minimum Steiner star for any finite point configuration in the plane or 3-space. The maximum ratio between the two is conjectured to be 4/π in the plane and 4/3 in three dimensions. Here we improve the upper bound to 1.3999 in the plane, and to √2 - 10-4 in 3-space. Our results also imply improved bounds on the maximum ratios between the minimum star and the maximum matching in two and three dimensions. Our method relies on constructing a suitable discretization for a continuous problem and then using linear programming to optimize over a relatively large set of constraints.