Topics in matrix analysis
Solving minimum-cost flow problems by successive approximation
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Matching is as easy as matrix inversion
Combinatorica
A polynomial-time algorithm, based on Newton's method, for linear programming
Mathematical Programming: Series A and B
A faster strongly polynomial minimum cost flow algorithm
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Finding minimum-cost flows by double scaling
Mathematical Programming: Series A and B
Iterative solution methods
Interior point algorithms: theory and analysis
Interior point algorithms: theory and analysis
Polynomial-time highest-gain augmenting path algorithms for the generalized circulation problem
Mathematics of Operations Research
Faster Algorithms for the Generalized Network Flow Problem
Mathematics of Operations Research
Beyond the flow decomposition barrier
Journal of the ACM (JACM)
On the Iterative Criterion for Generalized Diagonally Dominant Matrices
SIAM Journal on Matrix Analysis and Applications
Condition Numbers, the Barrier Method, and the Conjugate-Gradient Method
SIAM Journal on Optimization
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Speeding-up linear programming using fast matrix multiplication
SFCS '89 Proceedings of the 30th Annual Symposium on Foundations of Computer Science
Homology flows, cohomology cuts
Proceedings of the forty-first annual ACM symposium on Theory of computing
Hierarchical Diagonal Blocking and Precision Reduction Applied to Combinatorial Multigrid
Proceedings of the 2010 ACM/IEEE International Conference for High Performance Computing, Networking, Storage and Analysis
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
Computer Vision and Image Understanding
The laplacian paradigm: emerging algorithms for massive graphs
TAMC'10 Proceedings of the 7th annual conference on Theory and Applications of Models of Computation
A fast solver for a class of linear systems
Communications of the ACM
Runtime guarantees for regression problems
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
Parallel graph decompositions using random shifts
Proceedings of the twenty-fifth annual ACM symposium on Parallelism in algorithms and architectures
A new approach to computing maximum flows using electrical flows
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
A simple, combinatorial algorithm for solving SDD systems in nearly-linear time
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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We present asymptotically faster approximation algorithms for the generalized flow problems in which multipliers on edges are at most 1. For this lossy version of the maximum generalized flow problem, we obtain an additive ε approximation of the maximum flow in time O{m3/2 log (U/ε)2}, where m is the number of edges in the graph, all capacities are integers in the range {1, ... , U}, and all loss multipliers are ratios of integers in this range. For minimum cost lossy generalized flow with costs in the range {1,... ,U}, we obtain a flow that has value within an additive ε of the maximum value and cost at most the optimal cost. In many parameter ranges, these algorithms improve over the previously fastest algorithms for the generalized maximum flow problem by a factor of m1/2 and for the minimum cost generalized flow problem by a factor of approximately m1/2/ ε2. The algorithms work by accelerating traditional interior point algorithms by quickly solving the system of linear equations that arises in each step. The contributions of this paper are twofold. First, we analyze the performance of interior point algorithms with approximate linear system solvers. This analysis alone provides an algorithm for the standard minimum cost flow problem that runs in time Om3/2 log U}--an improvement of roughly O{n / m1/2} over previous algorithms. Second, we examine the linear equations that arise when using an interior point algorithm to solve generalized flow problems. We observe that these belong to the family of symmetric M-matrices, and we then develop Om-time algorithms for solving linear systems in these matrices. These algorithms reduce the problem of solving a linear system in a symmetric M-matrix to that of solving O{log n} linear systems in symmetric diagonally-dominant matrices, which we can do in time Om using the algorithm of Spielman and Teng. All of our algorithms operate on numbers of bit length at most O{log n U / ε}.