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

• Eugene L. Lawler
Chapter
Part of the International Centre for Mechanical Sciences book series (CISM, volume 135)

## Abstract

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.

## Keywords

Transit Time Binary Search Time Ratio Closed Path Directed Cycle
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag Wien 1972

## Authors and Affiliations

• Eugene L. Lawler
• 1
1. 1.Department of Electrical EngineeringUniversity of Colorado formally with the Polytechnic Institute of BrooklynUSA