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An approach for proving lower bounds: solution of Gilbert-Pollak's conjecture on Steiner ratio

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2 Author(s)
D. -Z. Du ; Dept. of Comput. Sci., Princeton Univ., NJ, USA ; F. K. Hwang

A family of finitely many continuous functions on a polytope X , namely {gi(x)}i∈I, is considered, and the problem of minimizing the function f(x)=maxi∈Igi( x) on X is treated. It is shown that if every g i(x) is a concave function, then the minimum value of f(x) is achieved at finitely many special points in X. As an application, a long-standing problem about Steiner minimum trees and minimum spanning trees is solved. In particular, if P is a set of n points on the Euclidean plane and Ls(P) and Lm( P) denote the lengths of a Steiner minimum tree and a minimum spanning tree on P, respectively, it is proved that, for any P, LS(P)⩾√3L m(P)/2, as conjectured by E.N. Gilbert and H.O. Pollak (1968)

Published in:

Foundations of Computer Science, 1990. Proceedings., 31st Annual Symposium on

Date of Conference:

22-24 Oct 1990