By Achim Bachem (auth.), Rudolf Henn, Bernhard Korte, Werner Oettli (eds.)
Read Online or Download Optimization and Operations Research: Proceedings of a Workshop Held at the University of Bonn, October 2–8, 1977 PDF
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Extra info for Optimization and Operations Research: Proceedings of a Workshop Held at the University of Bonn, October 2–8, 1977
Example text
1- S and i, j l: E d .. B21J i l: E I c. < 1- S i/j j which implies that I is a solution for the partition problem. 5. Results The results of section 3 and 4 are summarized in the tables polynomial solvable 2 I . 1 2 1m-dim. Euclid. I l: k 11-dim. Euclid. I l: k 11-dim. Euclid. I l: k 11-dim. Euclid. I max max dij N 1 lBvT l: dij l: d ij max dij The status of the 2 I· l: max d ij problem is not clear. All other noneuclidean problems belong to one of the classes PS or NPC. Another open question is which Euclidean problems are NP-complete.
Euclid. [ max max d .. lJ a~imilar The problem can be solved in way as the previous one. All we have to do is to replace the values d ilj by and to take the maximum operation instead of summation in the recursion formulas of the dynamic programming procedure. The problem 2 [ m-dim. Euclid. [L ~ I B ,d L d ij An 0 (Cfil)2 m - 1 ) algorithm for this problem is described in BOCK [1974J. Such an algorithm is polynomial for fixed m. The problem 2 [ • [ max max d .. lJ Let G = CV, E) the graph of the problem.
Euclid. I max max dij it is sufficient to show that each k-partitionning B1 , ••• , ~ can be transformed into a k-partitionning E1 , ••• , with the property given in the theorem and that this transformation will not increase the value of the objective function. Bk Without loss of generality we assume that B1 = {~, ••• , xr } is the set containing the smallest point. Let y ~ B1 a point of V which is contained in the convex hull of B1 • Then r y.. 1 v = 1. , E A Xv) < E AV d(x. ::. 1 v v• 1 1 V • 1 ~ ,j =:-1 1 J 1" J Thus, the value of the objective function will not increase if we add to B1 all points of V which are in the convex hull of B1 • In B2 , ••• , ~ we eliminate all points belonging to the convex hull of B1 and apply the same procedure to the nonempty sets derived from B2 , ••• , ~.
