Worst-case growth rates of some classical problems of combinatorial optimization
SIAM Journal on Computing
How long can a euclidean traveling salesman tour be?
SIAM Journal on Discrete Mathematics
Mathematics of Operations Research
Lower bounds for rectilinear Steiner trees in bounded space
Information Processing Letters
A priori inequalities for the Euclidean traveling salesman
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
Computational Aspects of VLSI
New results on the old k-opt algorithm for the TSP
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
Stochastic minimum spanning trees in euclidean spaces
Proceedings of the twenty-seventh annual symposium on Computational geometry
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We consider graphs such as the minimum spanning tree, minimum Steiner tree, minimum matching, and traveling salesman tour for n points in the d-dimensional unit cube. For each of these graphs, we show that the worst-case sum of the dth powers of edge lengths is O(log n). This is a consequence of a general “gap theorem”: for any subadditive geometric graph, either the worst-case sum of edge lengths is O(nd–1)/d) and the sum of dth powers is O(log n), or the sum of edge lengths is &OHgr;(n). We look more closely at some specific graphs: the worst-case sum of d powers is O(1) for minimum matching, but &OHgr;(log n) for traveling salesman tour, which answers a question of Snyder and Steele.