Nearly Maximum Flows in Nearly Linear Time

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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 O(α⁻²² 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+-optimal single-commodity flows undirected graphs in time m¹⁺ᵒ⁽¹⁾⁻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+-optimal-flows in time O(mϕ⁻¹).