Scaling algorithms for network problems
Journal of Computer and System Sciences
Constructing a perfect matching is in random NC
Combinatorica
Matrix multiplication via arithmetic progressions
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Fibonacci heaps and their uses in improved network optimization algorithms
Journal of the ACM (JACM)
Matching is as easy as matrix inversion
Combinatorica
A faster strongly polynomial minimum cost flow algorithm
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Almost-optimum speed-ups of algorithms for bipartite matching and related problems
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Faster scaling algorithms for network problems
SIAM Journal on Computing
Optimization
Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems
Journal of the ACM (JACM)
Parallelism in random access machines
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
Improved processor bounds for algebraic and combinatorial problems in RNC
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
Sublinear-time parallel algorithms for matching and related problems
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
A fast parallel algorithm for minimum-cost small integral flows
Euro-Par'12 Proceedings of the 18th international conference on Parallel Processing
A distributed multi-agent production planning and scheduling framework for mobile robots
Computers and Industrial Engineering
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Let G = (V, E) be a network for an assignment problem with 2n nodes and m edges, in which the largest edge cost is C. Recently the class of instances of bipartite matching problems has been shown to be in RNC provided that C is O(log^kn) for some fixed k. We show how to use scaling so as to develop an improved parallel algorithm and show that bipartite matching problems are in the class RNC provided that C = O(n^l^o^g^^^k^n) for some fixed k. We then generalize these results to minimum-cost flow problems. Let U be an upper bound on the capacities of the edges and on the largest demand. We show that the minimum-cost flow problems is in the class RNC, provided that log(C + U) = O(log^kn) for some fixed k. Thus the minimum-cost flow problem is in the class RNC even when the magnitude of the costs and capacities are allowed to grow faster than any polynomial in n. The key to our approach is to reduce the number of processors needed from an amount that is proportional to the magnitude of the largest edge cost to an amount that is independent of the magnitude of the largest edge cost. The tradeoff is an increase in the running time that grows linearly in log(C + U).