On the length of optimal TSP circuits in sets of bounded diameter

Abstract

Let V be a set of n points in Rk. Let d(V) denote the diameter of V, and l(V) denote the length of the shortest circuit which passes through all the points of V. (Such a circuit is an "optimal TSP circuit".) lk(n) are the extremal values of l(V) defined by lk(n)=max{l(V)|V∈Vnk}, where Vnk={V|V⊆Rk,|V|=n, d(V)=1}. A set V∈Vnk is "longest" if l(V)=lk(n). In this paper, first some geometrical properties of longest sets in R2 are studied which are used to obtain l2(n) for small n′s, and then asymptotic bounds on lk(n) are derived. Let δ(V) denote the minimal distance between a pair of points in V, and let: δk(n)=max{δ(V)|V∈Vnk}. It is easily observed that δk(n)=O(n-1 k). Hence, ck=lim supn→∞δk(n)n1 k exists. It is shown that for all n, ckn-1 k≤δk(n), and hence, for all n, lk(n)≥ ckn1-1 k. For k=2, this implies that l2(n)≥( π2 12)1 4n1 2, which generalizes an observation of Fejes-Toth that limn→∞l2(n)n-1 2≥( π2 12)1 4. It is also shown that lk(n) ≤ [ (3-√3)k (k-1)]nδk(n) + o(n1- 1 k) ≤ [ (3-√3)k (k-1)]n1- 1 k + o(n1- 1 k). The above upper bound is used to improve related results on longest sets in k-dimensional unit cubes obtained by Few (Mathematika 2 (1955), 141-144) for almost all k′s. For k=2, Few's technique is used to show that l2(n)≤( πn 2)1 2 + O(1). © 1984.