Computational experience with an approximation algorithm on large-scale Euclidean matching instances

We consider a 2-approximation algorithm for Euclidean minimum-cost perfect matching instances proposed by the authors in a previous paper. We present computational results for both random and real-world instances having between 1,000 and 131,072 vertices. The results indicate that our algorithm generates a matching within 2% of optimal in most cases. In over 1,400 experiments, the algorithm was never more than 4% from optimal. For the purposes of the study, we give a new implementation of the algorithm that uses linear space instead of quadratic space, and appears to run faster in practice.

[1]  Jon Louis Bentley,et al.  K-d trees for semidynamic point sets , 1990, SCG '90.

[2]  Eugene L. Lawler,et al.  The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization , 1985 .

[3]  Edward M. Reingold,et al.  Probabilistic analysis of divide-and-conquer heuristics for minimum weighted euclidean matching , 1983, Networks.

[4]  David S. Johnson,et al.  Data structures for traveling salesmen , 1993, SODA '93.

[5]  J. Steele,et al.  On the number of leaves of a euclidean minimal spanning tree , 1987, Journal of Applied Probability.

[6]  Kazuo Murota,et al.  Heuristics for planar minimum-weight perfect metchings , 1983, Networks.

[7]  Patrick Jaillet.  Rates of Convergence for Quasi-Additive Smooth Euclidean Functionals and Application to Combinatorial Optimization Problems , 1992, Math. Oper. Res..

[8]  J. Steele.  Subadditive Euclidean Functionals and Nonlinear Growth in Geometric Probability , 1981 .

[9]  J. Beardwood,et al.  The shortest path through many points , 1959, Mathematical Proceedings of the Cambridge Philosophical Society.

[10]  J. Michael Steele,et al.  Probabilistic and Worst Case Analyses of Classical Problems of Combinatorial Optimization in Euclidean Space , 1990, Math. Oper. Res..

[11]  Edward M. Reingold,et al.  Heuristics for weighted perfect matching , 1980, STOC '80.

[12]  Jon Louis Bentley,et al.  An Algorithm for Finding Best Matches in Logarithmic Expected Time , 1977, TOMS.

[13]  William J. Cook,et al.  Solving Large-Scale Matching Problems , 1991, Network Flows And Matching.

[14]  Gerhard Reinelt,et al.  TSPLIB - A Traveling Salesman Problem Library , 1991, INFORMS J. Comput..

[15]  J. Michael Steele,et al.  Complete Convergence of Short Paths and Karp's Algorithm for the TSP , 1981, Math. Oper. Res..

[16]  N. Biggs.  THE TRAVELING SALESMAN PROBLEM A Guided Tour of Combinatorial Optimization , 1986 .

[17]  Jon Louis Bentley,et al.  Fast Algorithms for Constructing Minimal Spanning Trees in Coordinate Spaces , 1978, IEEE Transactions on Computers.

[18]  David Avis,et al.  A survey of heuristics for the weighted matching problem , 1983, Networks.

[19]  Jon Jouis Bentley,et al.  Fast Algorithms for Geometric Traveling Salesman Problems , 1992, INFORMS J. Comput..

[20]  David P. Williamson,et al.  A general approximation technique for constrained forest problems , 1992, SODA '92.

[21]  Wansoo T. Rhee,et al.  A sharp deviation inequality for the stochastic traveling salesman problem , 1989 .