Nearly Maximum Flows in Nearly Linear Time

We introduce a new approach to the maximum flow problem in undirected, capacitated graphs using α-congestion-approximators: easy-to-compute functions that approximate the congestion required to route single-commodity demands in a graph to within a factor of α. Our algorithm maintains an arbitrary flow that may have some residual excess and deficits, while taking steps to minimize a potential function measuring the congestion of the current flow plus an over-estimate of the congestion required to route the residual demand. Since the residual term over-estimates, the descent process gradually moves the contribution to our potential function from the residual term to the congestion term, eventually achieving a flow routing the desired demands with nearly minimal congestion after tildeO(αeps-2log2 n) iterations. Our approach is similar in spirit to that used by Spielman and Teng (STOC 2004) for solving Laplacian systems, and we summarize our approach as trying to do for ℓ_∞-flows what they do for ℓ2-flows. Together with a nearly linear time construction of a no(1)-congestion-approximator, we obtain 1+eps-optimal single-commodity flows undirected graphs in time m1+o(1)eps-2, yielding the fastest known algorithm for that problem. Our requirements of a congestion-approximator are quite low, suggesting even faster and simpler algorithms for certain classes of graphs. For example, an α-competitive oblivious routing tree meets our definition, even without knowing how to route the tree back in the graph. For graphs of conductance φ, a trivial φ-1-congestion-approximator gives an extremely simple algorithm for finding 1+eps-optimal-flows in time tildeO(mφ-1).