35 research outputs found
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
Similarity-Aware Spectral Sparsification by Edge Filtering
In recent years, spectral graph sparsification techniques that can compute
ultra-sparse graph proxies have been extensively studied for accelerating
various numerical and graph-related applications. Prior nearly-linear-time
spectral sparsification methods first extract low-stretch spanning tree from
the original graph to form the backbone of the sparsifier, and then recover
small portions of spectrally-critical off-tree edges to the spanning tree to
significantly improve the approximation quality. However, it is not clear how
many off-tree edges should be recovered for achieving a desired spectral
similarity level within the sparsifier. Motivated by recent graph signal
processing techniques, this paper proposes a similarity-aware spectral graph
sparsification framework that leverages efficient spectral off-tree edge
embedding and filtering schemes to construct spectral sparsifiers with
guaranteed spectral similarity (relative condition number) level. An iterative
graph densification scheme is introduced to facilitate efficient and effective
filtering of off-tree edges for highly ill-conditioned problems. The proposed
method has been validated using various kinds of graphs obtained from public
domain sparse matrix collections relevant to VLSI CAD, finite element analysis,
as well as social and data networks frequently studied in many machine learning
and data mining applications
Oracle-Based Primal-Dual Algorithms for Packing and Covering Semidefinite Programs
Packing and covering semidefinite programs (SDPs) appear in natural relaxations of many combinatorial optimization problems as well as a number of other applications. Recently, several techniques were proposed, that utilize the particular structure of this class of problems, to obtain more efficient algorithms than those offered by general SDP solvers. For certain applications, such as those described in this paper, it maybe required to deal with SDP\u27s with exponentially or infinitely many constraints, which are accessible only via an oracle. In this paper, we give an efficient primal-dual algorithm to solve the problem in this case, which is an extension of a logarithmic-potential based algorithm of Grigoriadis, Khachiyan, Porkolab and Villavicencio (SIAM Journal of Optimization 41 (2001)) for packing/covering linear programs
Communication-Optimal Distributed Dynamic Graph Clustering
We consider the problem of clustering graph nodes over large-scale dynamic
graphs, such as citation networks, images and web networks, when graph updates
such as node/edge insertions/deletions are observed distributively. We propose
communication-efficient algorithms for two well-established communication
models namely the message passing and the blackboard models. Given a graph with
nodes that is observed at remote sites over time , the two
proposed algorithms have communication costs and
( hides a polylogarithmic factor), almost matching
their lower bounds, and , respectively, in the
message passing and the blackboard models. More importantly, we prove that at
each time point in our algorithms generate clustering quality nearly as
good as that of centralizing all updates up to that time and then applying a
standard centralized clustering algorithm. We conducted extensive experiments
on both synthetic and real-life datasets which confirmed the communication
efficiency of our approach over baseline algorithms while achieving comparable
clustering results.Comment: Accepted and to appear in AAAI'1