## Operations, Information and Decisions Papers

Journal Article

1995

#### Publication Source

SIAM Journal on Computing

24

3

665

671

#### DOI

10.1137/S0097539792226771

#### Abstract

It is proved that there are constants \$c_{1}\$, \$c_{2}\$, and \$c_{3}\$ such that for any set S of n points in the unit square and for any minimum-length tour T of S the sum of squares of the edge lengths of T is bounded by \$c_{1} \log n\$. (2) the number of edges having length t or greater in T is at most \$c_{2}/t^{2}\$, and (3) the sum of edge lengths of any subset E of T is bounded by \$c_{3}|E|^{1/2}\$. The second and third bounds are independent of the number of points in S, as well as their locations. Extensions to dimensions \$d > 2\$ are also sketched. The presence of the logarithmic term in (1) is engaging because such a term is not needed in the case of the minimum spanning tree and several analogous problems, and, furthermore, we know that there always exists some tour of S (which perhaps does not have minimal length) for which the sum of squared edges is bounded independently of n.

#### Keywords

Euclidean traveling salesman problem, inequalities, squared edge lengths, long edges

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Date Posted: 27 November 2017