Worst case asymptotics for some classical optimization problems

Abstract

We provide a simple and natural method for obtaining the worst case asymptotics of some of the classical problems in combinatorial optimization and operations research. Worst case asymptotics for the minimal spanning tree, shortest tour, and minimal matching onn points are found. The key simplifying idea involves consideration of the associated “boundary” processes. The general approach considered here also handles the case of power weighted edges.

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References

  1. [1]

    J. Beardwood, J. H. Halton, andJ. M. Hammersley: The shortest path through many points,Proc. Camb. Philos. Soc.,55 (1959), 299–327.

    Google Scholar 

  2. [2]

    C. Redmond, andJ. E. Yukich: Limit theorems and rates of convergence for Euclidean functionals,Ann. Appl. Prob.,4 (1994), 1057–1073.

    Google Scholar 

  3. [3]

    C. Redmond, andJ. E. Yukich: Asymptotics for Euclidean functionals with power weighted edges,Stoch. Proc. and Their Applications,6 (1996), 289–304.

    Google Scholar 

  4. [4]

    T. L. Snyder: Asymptotic worst case lengths in some problems from classical computational geometry and combinatorial optimization, Ph.D. thesis, Princeton University, Princeton NJ, 1987.

    Google Scholar 

  5. [5]

    T. L. Snyder, andJ. M. Steele: Worst-case greedy matchings in the unitd-cube,Networks,20 (1990), 779–800.

    Google Scholar 

  6. [6]

    J. M. Steele: Subadditive Euclidean functionals and non-linear growth in geometric probability,Ann. Prob.,9 (1981), 365–376.

    Google Scholar 

  7. [7]

    J. M. Steele: Seedlings in the theory of shortest paths, inDisorder in Physical systems: A volume in Honor of J. M. Hammersley, (G. Grimmett and D. Welsh, eds.) Cambridge University Press, 1990, London, 277–306.

    Google Scholar 

  8. [8]

    J. M. Steele, andT. L. Snyder: Worst-case growth rates of some classical problems of combinatorial optimization,Siam J. Computation,18 (1989), 278–287.

    Google Scholar 

  9. [9]

    J. E. Yukich: Asymptotics for the Euclidean TSP with power-weighted edges,Prob. Theory and Related Fields,102 (1995), 203–220.

    Google Scholar 

  10. [10]

    J. E. Yukich: Ergodic theorems for some classical problems of combinatorial optimizations,Ann. Appl. Orib. (1996), to appear.

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Research supported in part by NSA grant MDA904-95-H-1005.

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Yukich, J.E. Worst case asymptotics for some classical optimization problems. Combinatorica 16, 575–586 (1996). https://doi.org/10.1007/BF01271275

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Mathematics Subject Classification (1991)

  • 05 C 35
  • 90 C 10
  • 52 A 40