Timothy Law Snyder

Worst-case growth rates of some classical problems of combinatorial optimization

A method is presented for determining the asymptotic worst-case behavior of quantities like the length of the minimal spanning tree or the length of an optimal traveling salesman tour of n points in the unit d-cube. In each of these classical problems, the worst-case lengths are proved to have the exact asymptotic growth rate of

On the exact location of Steiner points in general dimension

\beta _n^{{{(d - 1)} / d}}

Convex hulls of random walks

, where

Lower bounds for rectilinear Steiner trees in bounded space

\beta

On minimal rectilinear Steiner trees in all dimensions

is a positive constant depending on the problem and the dimension. These results complement known results on the growth rates for the analogous quantities under probabilistic assumptions on the points, but the results given here are free of any probabilistic hypotheses.

A Priori Bounds on the Euclidean Traveling Salesman

The Steiner Problem is to form a minimum-length tree that contains a given set of points, where augmentation of the point set with additional (Steiner) points is permitted. The Rectilinear Steiner Problem is one in which edge weights are determined by the

A priori inequalities for the Euclidean traveling salesman

L_1

Worst-case minimum rectilinear steiner trees in all dimensions

, or Manhattan, distance between points in

Equidistribution in All Dimensions of Worst-Case Point Sets forthe Traveling Salesman Problem

\mathbb{R}^d

Worst-case greedy matchings in the unit d-cube

.Let S be a set of n points