2 research outputs found
A Linear-time Algorithm for Sparsification of Unweighted Graphs
Given an undirected graph and an error parameter , the {\em
graph sparsification} problem requires sampling edges in and giving the
sampled edges appropriate weights to obtain a sparse graph with
the following property: the weight of every cut in is within a
factor of of the weight of the corresponding cut in . If
is unweighted, an -time algorithm for constructing
with edges in expectation, and an
-time algorithm for constructing with edges in expectation have recently been developed
(Hariharan-Panigrahi, 2010). In this paper, we improve these results by giving
an -time algorithm for constructing with edges in expectation, for unweighted graphs. Our algorithm is
optimal in terms of its time complexity; further, no efficient algorithm is
known for constructing a sparser . Our algorithm is Monte-Carlo,
i.e. it produces the correct output with high probability, as are all efficient
graph sparsification algorithms
Sparse Sums of Positive Semidefinite Matrices
Recently there has been much interest in "sparsifying" sums of rank one
matrices: modifying the coefficients such that only a few are nonzero, while
approximately preserving the matrix that results from the sum. Results of this
sort have found applications in many different areas, including sparsifying
graphs. In this paper we consider the more general problem of sparsifying sums
of positive semidefinite matrices that have arbitrary rank.
We give several algorithms for solving this problem. The first algorithm is
based on the method of Batson, Spielman and Srivastava (2009). The second
algorithm is based on the matrix multiplicative weights update method of Arora
and Kale (2007). We also highlight an interesting connection between these two
algorithms.
Our algorithms have numerous applications. We show how they can be used to
construct graph sparsifiers with auxiliary constraints, sparsifiers of
hypergraphs, and sparse solutions to semidefinite programs