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Spectral Sparsification of Graphs
TLDR
It is proved that every graph has a spectral sparsifier of nearly linear size, and an algorithm is presented that produces spectralSparsifiers in time $O(m\log^{c}m)$, where $m$ is the number of edges in the original graph and $c$ is some absolute constant.
Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems
We present algorithms for solving symmetric, diagonally-dominant linear systems to accuracy ε in time linear in their number of non-zeros and log (κf (A) ε), where κf (A) is the condition number of
Settling the complexity of computing two-player Nash equilibria
We prove that Bimatrix, the problem of finding a Nash equilibrium in a two-player game, is complete for the complexity class PPAD (Polynomial Parity Argument, Directed version) introduced by
Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time
TLDR
The smoothed analysis of algorithms is introduced, which is a hybrid of the worst-case and average-case analysis of algorithm performance and shows that the shadow-vertex simplex algorithm has polynomial smoothed complexity.
On Trip Planning Queries in Spatial Databases
TLDR
This paper provides a numb er of approximation algorithms with approximation ratios that depend on either the number of categories, the maximum number of points per category or b oth, since they all run in polynomial time.
Smoothed Analysis of the Condition Numbers and Growth Factors of Matrices
TLDR
It is shown that the smoothed precision necessary to solve Ax = b, for any b, using Gaussian elimination without pivoting is logarithmic.
A Local Clustering Algorithm for Massive Graphs and Its Application to Nearly Linear Time Graph Partitioning
TLDR
This work presents a local clustering algorithm, a useful primitive for handling massive graphs, such as social networks and web-graphs, that finds a good cluster---a subset of vertices whose internal connections are significantly richer than its external connections---near a given vertex.
Nearly-Linear Time Algorithms for Preconditioning and Solving Symmetric, Diagonally Dominant Linear Systems
TLDR
A randomized algorithm is presented that on input a symmetric, weakly diagonally dominant matrix A with nonpositive off-diagonal entries and an n-vector produces an x such that $\tilde{x} - A^{\dagger} {b} \leq \epsilon$ in expected time.
Lower-stretch spanning trees
TLDR
A novel graph decomposition technique is used to improve the running time of the recent solver for symmetric diagonally dominant linear systems of Spielman and Teng and to improve several earlier approximation algorithms that use low-stretch spanning trees.
Local Computation of PageRank Contributions
TLDR
This work gives an efficient local algorithm that computes an ε-approximation of the contribution vector for a given vertex by adaptively examining O(1/ε) vertices and gives a local approximation algorithm for the primitive defined above.
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