AbstractThis paper presents a new approach to finding the minimum cuts in undirected graphs. The fundamental principle is simple: the edges in a graph's minimization cut form an extremely small fraction of the graph's edges. Using this idea, we give a randomized, strongly polynomial algorithm that finds the minimum cut in an arbitrarily weighted undirected graph with high probability. The algorithm runs in

O(n^2 log^3n) time, a significant improvement over the previous \tilde{O}(mn) time bounds based on maximum flows. It is simple and intuitive and used no complex data structures. Our algorithm can be parallelized to run in RNC withn^2 processors; this gives the first proof that the minimum cut problem can be solved in RNC. The algorithm does more than find a single minimum cut; it finds all of them.With minor modifications, our algorithm solves two other problems of interest. Our algorithm finds all cuts with value within a multiplicative factor of

aof the minimum cut's in expected \tilde{O}(n^{2a}) time, or in RNC withn^{2a} processors. The problem of finding a minimum multiway cut of a graph intorpieces is solved in expected \tilde{O}(n^{2(r-1)}) time, or in RNC withn^{2(r-1)} processors. The ``trace'' of the algorithm's execution on these two problems forms a new compact data structure for representing all small cuts and all multiway cuts in a graph. This data structure can be efficiently transformed into the more standard cactus representation for minimum cuts.The abstract is also available as a LaTeX file, a DVI file, or a PostScript file.

Categories and Subject Descriptors: F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems; G.2.2 [Discrete Mathematics]: Graph Theory --graph algorithms,network problems; G.3 [Probability and Statistics] --probabilistic algorithms (including Monte Carlo); I.1.2 [Algebraic Manipulation]: Algorithms

General Terms: Algorithms

Additional Key Words and Phrases: Graph algorithm, minimum cut, network reliability, parallel computing, randomized algorithm

Selected papers that cite this one

- András Benczúr and David R. Karger. Augmenting undirected edge connectivity in \tilde{
O}(n^2) time. In Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 500-509, San Francisco, California, 25-27 January 1998.

- David R. Karger and Ray P. Tai. Implementing a fully polynomial time approximation scheme for all terminal network reliability. In Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 334-343, New Orleans, Louisiana, 5-7 January 1997.

- Stavros G. Kolliopoulos and Clifford Stein. Finding real-valued single-source shortest paths in
o(n^3) expected time. Journal of Algorithms, 28(1):125-141, July 1998.

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