Optimal Cycles in Graphs and the Minimal Cost-To-Time Ratio Problem
Let G.= (N,A) be a directed graph, and let cij and t ij be a “cost” and a “transit time assigned to each arc (i, j). We consider three problems. The Negative Cycle Problem: Does G contain a directed cycle for which the sum of the costs is strictly negative? The Minimal-Cost Cycle Problem: As suming G contains no negative cycles, find a directed cycle for which the sum of the costs in minimum. The Minimal Cost-to-Time Ratio Cycle Problem: Find a directed cycle for which the sum of the costs divided by the sum of the transit times is minimum.
The first two problems can be solved by adaptations of well-know shortest path algorithms which require O (n3) computational steps, where n is the number of nodes of G. The minimal cost-to-time ratio problem can be solved by a procedure which uses a negative cycle algorithm as a subroutine. This procedure is essentially O (n3 log n) in complexity.
Various generalizations of these problems, in the form of side constraints on the cycles, are indicated.
KeywordsTransit Time Binary Search Time Ratio Closed Path Directed Cycle
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