Binomial random variate generation
Communications of the ACM
Computing edge-connectivity in multigraphs and capacitated graphs
SIAM Journal on Discrete Mathematics
Random sampling in cut, flow, and network design problems
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
Randomized algorithms
A new approach to the minimum cut problem
Journal of the ACM (JACM)
Approximating s-t minimum cuts in Õ(n2) time
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Global min-cuts in RNC, and other ramifications of a simple min-out algorithm
SODA '93 Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms
Random Sampling in Cut, Flow, and Network Design Problems
Mathematics of Operations Research
SIAM Journal on Computing
Random sampling in residual graphs
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Graph sparsification by effective resistances
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Expanders via random spanning trees
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Proceedings of the forty-first annual ACM symposium on Theory of computing
Graph partitioning using single commodity flows
Journal of the ACM (JACM)
Breaking the Multicommodity Flow Barrier for O(vlog n)-Approximations to Sparsest Cut
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
An O(log n/ log log n)-approximation algorithm for the asymmetric traveling salesman problem
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Edge Disjoint Paths in Moderately Connected Graphs
SIAM Journal on Computing
Approaching Optimality for Solving SDD Linear Systems
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Fast Approximation Algorithms for Cut-Based Problems in Undirected Graphs
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Sparsification of influence networks
Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
Spectral sparsification via random spanners
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
How user behavior is related to social affinity
Proceedings of the fifth ACM international conference on Web search and data mining
Graph sketches: sparsification, spanners, and subgraphs
PODS '12 Proceedings of the 31st symposium on Principles of Database Systems
Efficiently computing k-edge connected components via graph decomposition
Proceedings of the 2013 ACM SIGMOD International Conference on Management of Data
Spectral sparsification of graphs: theory and algorithms
Communications of the ACM
Proceedings of the International Conference on Computer-Aided Design
Hi-index | 0.02 |
We present a general framework for constructing cut sparsifiers in undirected graphs --- weighted subgraphs for which every cut has the same weight as the original graph, up to a multiplicative factor of (1 ε). Using this framework, we simplify, unify and improve upon previous sparsification results. As simple instantiations of this framework, we show that sparsifiers can be constructed by sampling edges according to their strength (a result of Benczur and Karger), effective resistance (a result of Spielman and Srivastava), edge connectivity, or by sampling random spanning trees. Sampling according to edge connectivity is the most aggressive method, and the most challenging to analyze. Our proof that this method produces sparsifiers resolves an open question of Benczur and Karger. While the above results are interesting from a combinatorial standpoint, we also prove new algorithmic results. In particular, we develop techniques that give the first (optimal) O(m)-time sparsification algorithm for unweighted graphs. Our algorithm has a running time of O(m) + ~O(n/ε2) for weighted graphs, which is also linear unless the input graph is very sparse itself. In both cases, this improves upon the previous best running times (due to Benczur and Karger) of O(m log2 n) (for the unweighted case) and O(m log3 n) (for the weighted case) respectively. Our algorithm constructs sparsifiers that contain O(n log n/ε2) edges in expectation; the only known construction of sparsifiers with fewer edges is by a substantially slower algorithm running in O(n3 m / ε2) time. A key ingredient of our proofs is a natural generalization of Karger's bound on the number of small cuts in an undirected graph. Given the numerous applications of Karger's bound, we suspect that our generalization will also be of independent interest.