An Algorithm Based on Graphs for Solving a Fair Division Problem

In this paper we propose an algorithm for solving a fair division problem: We want to find a distribution of a set of divisible items among a set of agents such that the agent who is getting the lowest amount obtains his maximum possible value. Even more, according to the found distribution, each agent obtains the same amount, and this is as high as possible. We model this situation as a linear programming problem and we use its dual problem for solving it. For doing this, we associate a bipartite graph with each set of dual variables. The algorithm presented uses the intrinsic relations in this kind of graphs for searching the optimal solution for the primal and dual problems. Finally, we show the convergence of the proposed algorithm.