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Extra info for Periodic Optimization: Volume I: Course Held at the Department of Automation and Information, June 1972
1) call for minimization over two alternatives. It is not clear that the first of these alternatives sired path does not pass through node native ( c~<;:> + c~)) (c<~:>) is that the da~ m and the second altar- is that it does. i- • We estimate the complexity of the Floyd-Warshall algorithm as follows. It is necessary to compute ~ • 1,2 ••.. 2, ... ,n and for c (m) .. ~. for m • 2,3, ... ,n+1. 1), requires one addition and one comparison. Thus, exactly n 3 additions and n 3 comparisons are required overall.
Edmonds, ''Path, Trees, and Flowers tt, Can. ]. , 17 (1965) 449-467.  M. Florian and P. Robert, "A Direct Search Method to Locate Negative Cycles in a Graph", Management Science, 17 (1971) 307-310. W. Floyd, nAlgorithm 97, Shortest Path", Comm. of the ACM; 5 (1962) 345. [s] L. R. , P-923 (Aug. 1965). L. Fox, 1 ~inding Minimal Cost-Time Ratio Circuits", Operations Research, 17 (1969) 546-551. L. Lawler, "Optimal Cycles in Doubly Weighted Directed Linear Graphs r;, Theory of Graphs, International Symposium, Dunod, Paris, and Gordon and Breach, New York (1966) 209-213.
5) As it is well known, such a problem can be described by means of a finite and oriented graph: every node represents a state and every arc ( ~· . ~4 i! 0 and q (k) constraint values \)' .. are associated to every arc ( ~ , 1 ) ~t from state ~ and the problem consists ;n the determination of the cycle with minimum cost per unit time. r) and '(:; S be the (r) = ~ ci.. "' , the loss rate (cost to time ratio) c (r) = is associated to every element ~(r) 't(l"") 1"" of S . 5). 5) are those in which only some of the constraints are p1·esent.