Graph Optimization Using Fractal Decomposition

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 worstcase 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 bounds on the optimal solution. We demonstrate this method on three example problems: all-pairs shortest path, all-pairs maximum flow, and cooperative search. Large-scale simulations show that this fractal decomposition method is computationally fast and can yield good results for practical problems.