Finding small simple cycle separators for 2-connected planar graphs
Journal of Computer and System Sciences
Edge separators of planar and outerplanar graphs with applications
Journal of Algorithms
Approximating s-t minimum cuts in Õ(n2) time
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Proof verification and the hardness of approximation problems
Journal of the ACM (JACM)
Property testing and its connection to learning and approximation
Journal of the ACM (JACM)
Small distortion and volume preserving embeddings for planar and Euclidean metrics
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Some optimal inapproximability results
Journal of the ACM (JACM)
On the optimality of the random hyperplane rounding technique for max cut
Random Structures & Algorithms - Probabilistic methods in combinatorial optimization
The regularity lemma and approximation schemes for dense problems
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Cuts, Trees and ℓ1-Embeddings of Graphs*
Combinatorica
Approximating the cut-norm via Grothendieck's inequality
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Lifts, Discrepancy and Nearly Optimal Spectral Gap
Combinatorica
Graph sparsification by effective resistances
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Dense Subsets of Pseudorandom Sets
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Column subset selection, matrix factorization, and eigenvalue optimization
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
An Efficient Sparse Regularity Concept
SIAM Journal on Discrete Mathematics
A general framework for graph sparsification
Proceedings of the forty-third annual ACM symposium on Theory of computing
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We initiate a principled study of graph densification. Given a graph G the goal of graph densification is to come up with another graph H that has significantly more edges than G but nevertheless approximates G well with respect to some set of test functions. In this paper we focus on the case of cut and spectral approximations. As it turns out graph densification exhibits rich connections to a set of interesting and sometimes seemingly unrelated questions in graph theory and metric embeddings. In particular we show the following results: • A graph G has a multiplicative cut approximation with an asymptotically increased density if and only if it does not embed into l1 under a weak notion of embeddability. We demonstrate that all planar graphs as well as random geometric graphs possess such embeddings and thus do not have densifiers. On the other hand, expanders do have densifiers (namely, the complete graph) and as a result do not embed into l1 even under our weak notion of embedding. • An analogous characterization is true for multiplicative spectral approximations where the embedding is into l22. Using this characterization we expose a surprisingly close connection between multiplicative spectral and multiplicative cut densifiers. • We also consider additive cut and spectral approximations. We exhibit graphs that do not possess non-trivial additive densifiers. Our results are mainly based on linear and semidefinite programs (and their duals) for computing the maximum weight densifier of a given graph. This also leads to efficient algorithms in the case of spectral densifiers and additive cut densifiers.