Graph Optimization Using Fractal Decomposition With Application to Cooperative Routing Problems

We introduce a method of hierarchically decomposing graph optimization problems to obtain approximate solutions with low computation. The method uses a partition on the graph to convert the original problem to a high level problem and several lower level problems. On each level, the resulting problems are in exactly the same form as the original one, so they can be further decomposed. In this way, the problems become fractal in nature. We use best-case and worst-case instances of the decomposed problems to establish upper and lower bounds on the optimal criteria, and these bounds are achieved with significantly less computation than what is required to solve the original problem. We show that as the number of hierarchical levels increases, the computational complexity approaches O(n) at the expense of looser approximation bounds. For regular lattice graphs, we provide constant factor bounds on the approximation error. We demonstrate the fractal decomposition method on three example problems related to cooperative routing: shortest path matrix, maximum flow matrix, and cooperative search. Large-scale simulations show that this fractal decomposition method is computationally fast and can yield good results for practical problems.