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A priori bounds on the Euclidean traveling salesman. (English) Zbl 0831.68077

Summary: 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 \(S(1)\) 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 becasue 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 \(n\).

MSC:

68R10 Graph theory (including graph drawing) in computer science
05C45 Eulerian and Hamiltonian graphs
90C35 Programming involving graphs or networks
68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
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