Learnability and the Vapnik-Chervonenkis dimension
Journal of the ACM (JACM)
On the optimal vertex-connectivity augmentation
Journal of Combinatorial Theory Series B
Centroids, representations, and submodular flows
SODA '93 Selected papers from the fourth annual ACM SIAM symposium on Discrete algorithms
Fast algorithms for k-shredders and k-node connectivity augmentation (extended abstract)
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Inferring tree topologies using flow tests
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Using expander graphs to find vertex connectivity
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Journal of Graph Theory
Semi-supervised learning using randomized mincuts
ICML '04 Proceedings of the twenty-first international conference on Machine learning
Finding small balanced separators
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
An analysis of graph cut size for transductive learning
ICML '06 Proceedings of the 23rd international conference on Machine learning
Detecting cuts in sensor networks
IPSN '05 Proceedings of the 4th international symposium on Information processing in sensor networks
Catching elephants with mice: sparse sampling for monitoring sensor networks
Proceedings of the 5th international conference on Embedded networked sensor systems
Detecting cuts in sensor networks
ACM Transactions on Sensor Networks (TOSN)
Catching elephants with mice: Sparse sampling for monitoring sensor networks
ACM Transactions on Sensor Networks (TOSN)
Degree Bounded Network Design with Metric Costs
SIAM Journal on Computing
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We consider a model for monitoring the connectivity of a network subject to node or edge failures. In particular, we are concerned with detecting (ε, k)-failures: events in which an adversary deletes up to network elements (nodes or edges), after which there are two sets of nodes A and B, each at least an ε fraction of the network, that are disconnected from one another. We say that a set D of nodes is an (ε k)-detection set if, for any (ε k)-failure of the network, some two nodes in D are no longer able to communicate; in this way, D "witnesses" any such failure. Recent results show that for any graph G, there is an is (ε k)-detection set of size bounded by a polynomial in k and ε, independent of the size of G.In this paper, we expose some relationships between bounds on detection sets and the edge-connectivity λ and node-connectivity κ of the underlying graph. Specifically, we show that detection set bounds can be made considerably stronger when parameterized by these connectivity values. We show that for an adversary that can delete κλ edges, there is always a detection set of size O((κ/ε) log (1/ε)) which can be found by random sampling. Moreover, an (ε, &lambda)-detection set of minimum size (which is at most 1/ε) can be computed in polynomial time. A crucial point is that these bounds are independent not just of the size of G but also of the value of λ.Extending these bounds to node failures is much more challenging. The most technically difficult result of this paper is that a random sample of O((κ/ε) log (1/ε)) nodes is a detection set for adversaries that can delete a number of nodes up to κ, the node-connectivity.For the case of edge-failures we use VC-dimension techniques and the cactus representation of all minimum edge-cuts of a graph; for node failures, we develop a novel approach for working with the much more complex set of all minimum node-cuts of a graph.