ScholarlyCommons

Title

Complete Convergence of Short Paths and Karp's Algorithm for the TSP

Author(s)

John M. Steele, University of Pennsylvania

Document Type

Journal Article

Date of this Version

1981

Publication Source

Mathematics of Operations Research

Volume

6

Issue

6

Start Page

374

Last Page

378

DOI

10.1287/moor.6.3.374

Abstract

Let X_i, 1 ≤ i < ∞, be uniformly distributed in [0, 1]² and let T_n be the length of the shortest closed path connecting {X₁, X₂, …, X_n}. It is proved that there is a constant 0 < β < ∞ such that for all ϵ > 0 ∑n=1∞p(|Tn/n−β‾‾‾‾‾√|>ϵ)<∞.n1∞pTnnβϵ∞

This result is essential in justifying Karp's algorithm for the traveling salesman problem under the independent model, and it settles a question posed by B. W. Weide.

Keywords

traveling salesman problem, complete convergence, sub additive processes, subadditive Euclidean functionals, jackknife, Efron-Stein inequality

Recommended Citation

Steele, J. M. (1981). Complete Convergence of Short Paths and Karp's Algorithm for the TSP. Mathematics of Operations Research, 6 (6), 374-378. http://dx.doi.org/10.1287/moor.6.3.374