A data structure for dynamic trees
Journal of Computer and System Sciences
Beyond the flow decomposition barrier
Journal of the ACM (JACM)
Better random sampling algorithms for flows in undirected graphs
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
Random sampling in residual graphs
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Proceedings of the 7th International IPCO Conference on Integer Programming and Combinatorial Optimization
Recent Developments in Maximum Flow Algorithms (Invited Lecture)
SWAT '98 Proceedings of the 6th Scandinavian Workshop on Algorithm Theory
Solving fractional packing problems in Oast(1/ε) iterations
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Smooth minimization of non-smooth functions
Mathematical Programming: Series A and B
Faster approximate lossy generalized flow via interior point algorithms
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Smooth Optimization with Approximate Gradient
SIAM Journal on Optimization
Perfect matchings in o(n log n) time in regular bipartite graphs
Proceedings of the forty-second ACM symposium on Theory of computing
Electrical flows, laplacian systems, and faster approximation of maximum flow in undirected graphs
Proceedings of the forty-third annual ACM symposium on Theory of computing
A Nearly-m log n Time Solver for SDD Linear Systems
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
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We give an algorithm which computes a (1-ε)-approximately maximum st-flow in an undirected uncapacitated graph in time O(1/ε√m/F⋅ m log2 n) where F is the flow value. By trading this off against the Karger-Levine algorithm for undirected graphs which takes ~O(m+nF) time, we obtain a running time of ~O(m n1/3/ε2/3) for uncapacitated graphs, improving the previous best dependence on ε by a factor of O(1/ε3). Like the algorithm of Christiano, Kelner, Madry, Spielman and Teng, our algorithm reduces the problem to electrical flow computations which are carried out in linear time using fast Laplacian solvers. However, in contrast to previous work, our algorithm does not reweight the edges of the graph in any way, and instead uses local (i.e., non s-t) electrical flows to reroute the flow on congested edges. The algorithm is simple and may be viewed as trying to find a point at the intersection of two convex sets (the affine subspace of st-flows of value F and the l∞ ball) by an accelerated version of the method of alternating projections due to Nesterov. By combining this with Ford and Fulkerson's augmenting paths algorithm, we obtain an exact algorithm with running time ~O(m5/4 F1/4) for uncapacitated undirected graphs, improving the previous best running time of ~O(m+ min(nF,m3/2)). We give a related algorithm with the same running time for approximate minimum cut, based on minimizing a smoothed version of the l1 norm inside the cut space of the input graph. We show that the minimizer of this norm is related to an approximate blocking flow and use this to give an algorithm for computing a length k approximately blocking flow in time ~O(m √k).