GRAPH SPARSIFICATION BY UNIVERSAL GREEDY ALGORITHMS  

作　　者：Ming-Jun Lai Jiaxin Xie Zhiqiang Xu 

机构地区：[1]Department of Mathematics,University of Georgia,Athens,GA 30602,USA [2]LMIB of the Ministry of Education,School of Mathematical Sciences,Beihang University,Beijing 100191,China [3]LSEC,Inst.Comp.Math.,Academy of Mathematics and System Science,Chinese Academy of Sciences,Beijing 100091,China [4]School of Mathematical Sciences,University of Chinese Academy of Sciences,Beijing 100049,China

出　　处：《Journal of Computational Mathematics》2023年第4期741-770,共30页计算数学（英文）

基　　金：supported by NSFC grant(Nos.12001026,12071019);supported by the National Science Fund for Distinguished Young Scholars grant(No.12025108);Beijing Natural Science Foundation(No.Z180002);NSFC grant(Nos.12021001,11688101).

摘　　要：Graph sparsification is to approximate an arbitrary graph by a sparse graph and is useful in many applications,such as simplification of social networks,least squares problems,and numerical solution of symmetric positive definite linear systems.In this paper,inspired by the well-known sparse signal recovery algorithm called orthogonal matching pursuit(OMP),we introduce a deterministic,greedy edge selection algorithm,which is called the universal greedy approach(UGA)for the graph sparsification problem.For a general spectral sparsification problem,e.g.,the positive subset selection problem from a set of m vectors in R n,we propose a nonnegative UGA algorithm which needs O(mn^(2)+n^(3)/ϵ^(2))time to find a 1+ϵ/β/1-ϵ/β-spectral sparsifier with positive coefficients with sparsity at most[n/ϵ^(2)],where β is the ratio between the smallest length and largest length of the vectors.The convergence of the nonnegative UGA algorithm is established.For the graph sparsification problem,another UGA algorithm is proposed which can output a 1+O(ϵ)/1-O(ϵ)-spectral sparsifier with[n/ϵ^(2)]edges in O(m+n^(2)/ϵ^(2))time from a graph with m edges and n vertices under some mild assumptions.This is a linear time algorithm in terms of the number of edges that the community of graph sparsification is looking for.The best result in the literature to the knowledge of the authors is the existence of a deterministic algorithm which is almost linear,i.e.O(m^(1+o(1)))for some o(1)=O((log log(m))^(2/3)/log^(1/3)(m)).Finally,extensive experimental results,including applications to graph clustering and least squares regression,show the effectiveness of proposed approaches.

关 键 词：Spectral sparsification Subset selection Greedy algorithms Graph clustering Linear sketching 

分 类 号：O157.5[理学—数学]