The Annals of Applied Probability

Ergodic theorems for some classical problems in combinatorial optimization

J. E. Yukich

Full-text: Open access

Abstract

We show that the stochastic versions of some classical problems in combinatorial optimization may be imbedded in multiparameter subadditive processes having an intrinsic ergodic structure. A multiparameter generalization of Kingman's subadditive ergodic theorem is used to capture strong laws for these optimization problems, including the traveling salesman and minimal spanning tree processes. In this way we make progress on some open problems and provide alternate proofs of some well known asymptotic results.

Article information

Source
Ann. Appl. Probab., Volume 6, Number 3 (1996), 1006-1023.

Dates
First available in Project Euclid: 18 October 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1034968238

Digital Object Identifier
doi:10.1214/aoap/1034968238

Mathematical Reviews number (MathSciNet)
MR1410126

Zentralblatt MATH identifier
0866.60027

Subjects
Primary: 60F15: Strong theorems 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60C05: Combinatorial probability

Keywords
Subadditive ergodic theorems combinatorial optimization minimal spanning tree traveling salesman problem boundary process

Citation

Yukich, J. E. Ergodic theorems for some classical problems in combinatorial optimization. Ann. Appl. Probab. 6 (1996), no. 3, 1006--1023. doi:10.1214/aoap/1034968238. https://projecteuclid.org/euclid.aoap/1034968238


Export citation

References

  • AKCOGLU, M. A. and KRENGEL, U. 1981. Ergodic theorems for superadditive processes. J. Reine Angew. Math. 323 53 67. Z.
  • ALDOUS, D. and STEELE, J. M. 1992. Asy mptotics for Euclidean minimal spanning trees on random points. Probab. Theory Related Fields 92 247 258. Z.
  • AVRAM, F. and BERTSIMAS, D. 1992. The minimum spanning tree constant in geometric probability and under the independent model: a unified approach. Ann. Appl. Probab. 2 113 130. Z.
  • BEARDWOOD, J., HALTON, J. H. and HAMMERSLEY, J. M. 1959. The shortest path through many points. Proceedings of Cambridge Philosophical Society 55 299 327. Z.
  • HOCHBAUM, D. and STEELE, J. M. 1982. Steinhaus's geometric location problem for random samples in the plane. Adv. in Appl. Probab. 14 55 67. Z.
  • KINGMAN, J. F. C. 1968. The ergodic theory of subadditive stochastic processes. J. Roy. Statist. Soc. Ser. B 30 499 510. Z.
  • KRENGEL, U. and Py KE, R. 1987. Uniform pointwise ergodic theorems for classes of averaging sets and multi-parameter processes. Stochastic Process. Appl. 26 289 296. Z.
  • REDMOND, C. and YUKICH, J. E. 1994. Limit theorems and rates of convergence for Euclidean functionals. Ann. Appl. Probab. 4 1057 1073. Z.
  • REDMOND, C. and YUKICH, J. E. 1996. Asy mptotics for Euclidean functionals with power weighted edges. Stochastic Process. Appl. 61 289 304. Z.
  • RHEE, W. S. 1993. A matching problem and subadditive Euclidean functionals. Ann. Appl. Probab. 3 794 801. Z.
  • SMy THE, R. T. 1976. Multiparameter subadditive processes. Ann. Probab. 4 772 782. Z.
  • STEELE, J. M. 1981. Subadditive Euclidean functionals and non-linear growth in geometric probability. Ann. Probab. 9 365 376. Z.
  • STEELE, J. M. 1988. Growth rates of Euclidean minimal spanning trees with power weighted edges. Ann. Probab. 16 1767 1787. Z.
  • STEELE, J. M. 1990. Seedlings in the theory of shortest paths. In Disorder in physical Sy stems: Z. A Volume in Honor of J. M. Hammersley G. Grimmett and D. Welsh, eds. 277 306. Cambridge Univ. Press. Z.
  • STEELE, J. M. 1992. Euclidean semi-matchings of random samples. Mathematical Programming 3 127 146. Z.
  • TALAGRAND, M. 1995. Concentration of measure and isoperimetric inequalities in product ´ spaces. Inst. Hautes Etudes Sci. Publ. Math. 81 73 205. Z.
  • TALAGRAND, M. 1996a. A new look at independence. Ann. Probab. 24 1 34. Z.
  • TALAGRAND, M. 1996b. New concentration inequalities in product spaces. Preprint. Z.
  • VAN ENTER, A. C. D. and VAN HEMMEN, J. L. 1983. The thermody namic limit for long-range random sy stems. J. Statist. Phy s. 32 141 152. Z.
  • YUKICH, J. E. 1995. Asy mptotics for the Euclidean TSP with power weighted edges. Probab. Theory Related Fields 102 203 220.
  • BETHLEHEM, PENNSy LVANIA 18015 E-MAIL: jey 0@lehigh.edu