407 research outputs found
Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time
We present the first almost-linear time algorithm for constructing
linear-sized spectral sparsification for graphs. This improves all previous
constructions of linear-sized spectral sparsification, which requires
time.
A key ingredient in our algorithm is a novel combination of two techniques
used in literature for constructing spectral sparsification: Random sampling by
effective resistance, and adaptive constructions based on barrier functions.Comment: 22 pages. A preliminary version of this paper is to appear in
proceedings of the 56th Annual IEEE Symposium on Foundations of Computer
Science (FOCS 2015
An SDP-Based Algorithm for Linear-Sized Spectral Sparsification
For any undirected and weighted graph with vertices and
edges, we call a sparse subgraph of , with proper reweighting of the
edges, a -spectral sparsifier if holds for any , where and
are the respective Laplacian matrices of and . Noticing that
time is needed for any algorithm to construct a spectral sparsifier and a
spectral sparsifier of requires edges, a natural question is to
investigate, for any constant , if a -spectral
sparsifier of with edges can be constructed in time,
where the notation suppresses polylogarithmic factors. All previous
constructions on spectral sparsification require either super-linear number of
edges or time.
In this work we answer this question affirmatively by presenting an algorithm
that, for any undirected graph and , outputs a
-spectral sparsifier of with edges in
time. Our algorithm is based on three novel
techniques: (1) a new potential function which is much easier to compute yet
has similar guarantees as the potential functions used in previous references;
(2) an efficient reduction from a two-sided spectral sparsifier to a one-sided
spectral sparsifier; (3) constructing a one-sided spectral sparsifier by a
semi-definite program.Comment: To appear at STOC'1
Augmenting the algebraic connectivity of graphs
For any undirected graph G=(V,E) and a set EW of candidate edges with E∩EW=∅, the (k,γ)-spectral augmentability problem is to find a set F of k edges from EW with appropriate weighting, such that the algebraic connectivity of the resulting graph H=(V,E∪F) is least γ. Because of a tight connection between the algebraic connectivity and many other graph parameters, including the graph's conductance and the mixing time of random walks in a graph, maximising the resulting graph's algebraic connectivity by adding a small number of edges has been studied over the past 15 years.
In this work we present an approximate and efficient algorithm for the (k,γ)-spectral augmentability problem, and our algorithm runs in almost-linear time under a wide regime of parameters. Our main algorithm is based on the following two novel techniques developed in the paper, which might have applications beyond the (k,γ)-spectral augmentability problem.
(1) We present a fast algorithm for solving a feasibility version of an SDP for the algebraic connectivity maximisation problem from [GB06]. Our algorithm is based on the classic primal-dual framework for solving SDP, which in turn uses the multiplicative weight update algorithm. We present a novel approach of unifying SDP constraints of different matrix and vector variables and give a good separation oracle accordingly.
(2) We present an efficient algorithm for the subgraph sparsification problem, and for a wide range of parameters our algorithm runs in almost-linear time, in contrast to the previously best known algorithm running in at least Ω(n2mk) time [KMST10]. Our analysis shows how the randomised BSS framework can be generalised in the setting of subgraph sparsification, and how the potential functions can be applied to approximately keep track of different subspaces
- …