TY - GEN

T1 - Faster sublinear approximation of the number of k-cliques in low-arboricity graphs

AU - Eden, Talya

AU - Ron, Dana

AU - Seshadhri, C.

N1 - Publisher Copyright:
Copyright © 2020 by SIAM

PY - 2020

Y1 - 2020

N2 - Given query access to an undirected graph G, we consider the problem of computing a (1 ± ε)-approximation of the number of k-cliques in G. The standard query model for general graphs allows for degree queries, neighbor queries, and pair queries. Let n be the number of vertices, m be the number of edges, and nk be the number of k-cliques. Previous work by Eden, Ron and Seshadhri (STOC 2018) n gives an O∗(n1/k + mk/2 )-time algorithm for this problem nk k (we use O∗(·) to suppress poly(log n, 1/ε, kk) dependencies). Moreover, this bound is nearly optimal when the expression is sublinear in the size of the graph. Our motivation is to circumvent this lower bound, by parameterizing the complexity in terms of graph arboricity. The arboricity of G is a measure for the graph density “everywhere”. There is a very rich family of graphs with bounded arboricity, including all minor-closed graph classes (such as planar graphs and graphs with bounded treewidth), bounded degree graphs, preferential attachment graphs and more. We design an algorithm for the class of graphs with arboricity at most α, whose running time is + mαk−2 n O∗(min{nαnkk−1 , }). We also prove a nearly n1/k nk k matching lower bound. For all graphs, the arboricity is O(√m), so this bound subsumes all previous results on sublinear clique approximation. As a special case of interest, consider minor-closed families of graphs, which have constant arboricity. Our result implies that for any minor-closed family of graphs, there is a (1 ± ε)-approximation algorithm for nk that has running time O∗(nnk ). Such a bound was not known even for the special (classic) case of triangle counting in planar graphs.

AB - Given query access to an undirected graph G, we consider the problem of computing a (1 ± ε)-approximation of the number of k-cliques in G. The standard query model for general graphs allows for degree queries, neighbor queries, and pair queries. Let n be the number of vertices, m be the number of edges, and nk be the number of k-cliques. Previous work by Eden, Ron and Seshadhri (STOC 2018) n gives an O∗(n1/k + mk/2 )-time algorithm for this problem nk k (we use O∗(·) to suppress poly(log n, 1/ε, kk) dependencies). Moreover, this bound is nearly optimal when the expression is sublinear in the size of the graph. Our motivation is to circumvent this lower bound, by parameterizing the complexity in terms of graph arboricity. The arboricity of G is a measure for the graph density “everywhere”. There is a very rich family of graphs with bounded arboricity, including all minor-closed graph classes (such as planar graphs and graphs with bounded treewidth), bounded degree graphs, preferential attachment graphs and more. We design an algorithm for the class of graphs with arboricity at most α, whose running time is + mαk−2 n O∗(min{nαnkk−1 , }). We also prove a nearly n1/k nk k matching lower bound. For all graphs, the arboricity is O(√m), so this bound subsumes all previous results on sublinear clique approximation. As a special case of interest, consider minor-closed families of graphs, which have constant arboricity. Our result implies that for any minor-closed family of graphs, there is a (1 ± ε)-approximation algorithm for nk that has running time O∗(nnk ). Such a bound was not known even for the special (classic) case of triangle counting in planar graphs.

UR - http://www.scopus.com/inward/record.url?scp=85084087932&partnerID=8YFLogxK

M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.conference???

AN - SCOPUS:85084087932

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 1467

EP - 1478

BT - 31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020

A2 - Chawla, Shuchi

PB - Association for Computing Machinery

Y2 - 5 January 2020 through 8 January 2020

ER -