Different Optimal Solutions in Shared Path Graphs

We examine an expansion upon the basic shortest path in graphs problem. We define journeys to be source-destination pairs in weighted and connected graphs, and allow them to equally split the cost of shared edges. In this new problem, there are multiple possible definitions of optimality. We investigate three: minimizing the total resources—the sum of the journeys’ costs— of a graph’s journeys; minimizing individual journeys’ costs using analysis from game theory with an aim of stable formations called Strong Nash Equilibria; and minimizing the maximum cost that any journey in a graph has to pay, a cooperative solution. We developed heuristics that, given any weighted, connected graph and a set of journeys, can manipulate the journeys into routes that approach these definitions of optimal. Two versions, speedy and exhaustive, were developed of the Strong Nash Equilibrium heuristic. Results showed that the speedy version was equally as effective as the exhaustive version 99.0% of the time. 18% of the tests on the cooperative heuristic gave different results from different initial conditions, indicating potential room for improvement.