Periodic Optimization pp 37-60 | Cite as

# Optimal Cycles in Graphs and the Minimal Cost-To-Time Ratio Problem

## Abstract

Let G.= (N,A) be a directed graph, and let c_{ij} 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 (n^{3}) 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 (n^{3} log n) in complexity.

Various generalizations of these problems, in the form of side constraints on the cycles, are indicated.

## Keywords

Transit Time Binary Search Time Ratio Closed Path Directed Cycle## Preview

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