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

1. iBtoodwtioB. The main objective of the present note is to solve a problem prĉ >osed by Weide (1978) coQceming the complete convergence of c^lain random variables associated with Karp's probabilistic analysis of the traveling salesman problem (Karp (1976), (1977)). To set the problem precisely let A',, 1 < i < oo, be ind^>eiMi(mt random variables uniformly distributed on the unit square [0, if, and let T„ d^iote the teagth of the shortest cl<»ed path (in the usual Euclidean distance) which connects each deawnt of {Xi,X2, . . ., X^}. It was proved by Beardwood, Halton, and Hammersley (1959) that