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This article summarizes the current status of several streams of research that deal with the probability theory of problems of combinatorial optimization. There is a particular emphasis on functionals of finite point sets. The most famous example of such functionals is the length associated with the Euclidean traveling salesman problem (TSP), but closely related problems include the minimal spanning tree problem, minimal matching problems and others. Progress is also surveyed on (1) the approximation and determination of constants whose existence is known by subadditive methods, (2) the central limit problems for several functionals closely related to Euclidean functionals, and (3) analogies in the asymptotic behavior between worst-case and expected-case behavior of Euclidean problems. No attempt has been made in this survey to cover the many important applications of probability to linear programming, arrangement searching or other problems that focus on lines or planes.
Matching, minimal spanning trees, steiner trees, subadditive euclidean functionals, traveling salesman problem, worst-case analyses
Steele, J. M. (1993). Probability and Problems in Euclidean Combinatorial Optimization. Statistical Science, 8 (1), 48-56. http://dx.doi.org/10.1214/ss/1177011083
Date Posted: 27 November 2017
This document has been peer reviewed.