Finding maximum flows in undirected graphs seems easier than bipartite matching
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Better random sampling algorithms for flows in undirected graphs
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
Minimum cuts in near-linear time
Journal of the ACM (JACM)
Random sampling in residual graphs
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision
IEEE Transactions on Pattern Analysis and Machine Intelligence
Approximating the smallest k-edge connected spanning subgraph by LP-rounding
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Guided design search in the interval-bounded sailor assignment problem
Computers and Operations Research
Capacity analysis of maximal flow in ad hoc networks
Proceedings of the 2006 international conference on Wireless communications and mobile computing
An analysis of graph cut size for transductive learning
ICML '06 Proceedings of the 23rd international conference on Machine learning
On unbiased sampling for unstructured peer-to-peer networks
Proceedings of the 6th ACM SIGCOMM conference on Internet measurement
On unbiased sampling for unstructured peer-to-peer networks
IEEE/ACM Transactions on Networking (TON)
Guided design search in the interval-bounded sailor assignment problem
Computers and Operations Research
GConnect: a connectivity index for massive disk-resident graphs
Proceedings of the VLDB Endowment
Sparse reliable graph backbones
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming: Part II
Edge Disjoint Paths in Moderately Connected Graphs
SIAM Journal on Computing
A general framework for graph sparsification
Proceedings of the forty-third annual ACM symposium on Theory of computing
Sparse reliable graph backbones
Information and Computation
On the communication and streaming complexity of maximum bipartite matching
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Multicriteria global minimum cuts
ISAAC'04 Proceedings of the 15th international conference on Algorithms and Computation
Approximation algorithms and hardness of integral concurrent flow
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Ranking and sparsifying a connection graph
WAW'12 Proceedings of the 9th international conference on Algorithms and Models for the Web Graph
Efficient community detection in large networks using content and links
Proceedings of the 22nd international conference on World Wide Web
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We use random sampling as a tool for solving undirected graph problems. We show that the sparse graph, or skeleton, that arises when we randomly sample a graph's edges will accurately approximate the value of all cuts in the original graph with high probability. This makes sampling effective for problems involving cuts in graphs. We present fast randomized (Monte Carlo and Las Vegas) algorithms for approximating and exactly finding minimum cuts and maximum flows in unweighted, undirected graphs. Our cut-approximation algorithms extend unchanged to weighted graphs while our weighted-graph flow algorithms are somewhat slower. Our approach gives a general paradigm with potential applications to any packing problem. It has since been used in a near-linear time algorithm for finding minimum cuts, as well as faster cut and flow algorithms. Our sampling theorems also yield faster algorithms for several other cut-based problems, including approximating the best balanced cut of a graph, finding a k-connected orientation of a 2k-connected graph, and finding integral multicommodity flows in graphs with a great deal of excess capacity. Our methods also improve the efficiency of some parallel cut and flow algorithms. Our methods also apply to the network design problem, where we wish to build a network satisfying certain connectivity requirements between vertices. We can purchase edges of various costs and wish to satisfy the requirements at minimum total cost. Since our sampling theorems apply even when the sampling probabilities are different for different edges, we can apply randomized rounding to solve network design problems. This gives approximation algorithms that guarantee much better approximations than previous algorithms whenever the minimum connectivity requirement is large. As a particular example, we improve the best approximation bound for the minimum k-connected subgraph problem from 1.85 to [math not displayed].