TY - GEN

T1 - Testing bounded arboricity

AU - Eden, Talya

AU - Levi, Reut

AU - Ron, Dana

N1 - Publisher Copyright: © Copyright 2018 by SIAM.

PY - 2018

Y1 - 2018

N2 - In this paper we consider the problem of testing whether a graph has bounded arboricity. The family of graphs with bounded arboricity includes, among others, bounded-degree graphs, all minor-closed graph classes (e.g. planar graphs, graphs with bounded treewidth) and randomly generated preferential attachment graphs. Graphs with bounded arboricity have been studied extensively in the past, in particular since for many problems they allow for much more efficient algorithms and/or better approximation ratios. We present a tolerant tester in the sparse-graphs model. The sparse-graphs model allows access to degree queries and neighbor queries, and the distance is defined with respect to the actual number of edges. More specifically, our algorithm distinguishes between graphs that are ϵ-close to having arboricity ϵand graphs that cí-far from having arboricity 3ϵ, where c is an absolute small constant. The query complexity and running time of the algorithm are1 Õ( n/√m · log(1/ϵ)/epsi; + n· α/m O(log(1/ϵ))where n denotes the number of vertices and m denotes the number of edges. In terms of the dependence on n and m this bound is optimal up to poly-logarithmic factors since (n= p m) queries are necessary (and the arboricity of a graph is always O( p m)). We leave it as an open question whether the dependence on 1/ϵ can be improved from quasi-polynomial to polynomial. Our techniques include an efficient local simulation for approximating the outcome of a global (almost) forestdecomposition algorithm as well as a tailored procedure of edge sampling.

AB - In this paper we consider the problem of testing whether a graph has bounded arboricity. The family of graphs with bounded arboricity includes, among others, bounded-degree graphs, all minor-closed graph classes (e.g. planar graphs, graphs with bounded treewidth) and randomly generated preferential attachment graphs. Graphs with bounded arboricity have been studied extensively in the past, in particular since for many problems they allow for much more efficient algorithms and/or better approximation ratios. We present a tolerant tester in the sparse-graphs model. The sparse-graphs model allows access to degree queries and neighbor queries, and the distance is defined with respect to the actual number of edges. More specifically, our algorithm distinguishes between graphs that are ϵ-close to having arboricity ϵand graphs that cí-far from having arboricity 3ϵ, where c is an absolute small constant. The query complexity and running time of the algorithm are1 Õ( n/√m · log(1/ϵ)/epsi; + n· α/m O(log(1/ϵ))where n denotes the number of vertices and m denotes the number of edges. In terms of the dependence on n and m this bound is optimal up to poly-logarithmic factors since (n= p m) queries are necessary (and the arboricity of a graph is always O( p m)). We leave it as an open question whether the dependence on 1/ϵ can be improved from quasi-polynomial to polynomial. Our techniques include an efficient local simulation for approximating the outcome of a global (almost) forestdecomposition algorithm as well as a tailored procedure of edge sampling.

UR - http://www.scopus.com/inward/record.url?scp=85045567263&partnerID=8YFLogxK

U2 - https://doi.org/10.1137/1.9781611975031.136

DO - https://doi.org/10.1137/1.9781611975031.136

M3 - منشور من مؤتمر

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

SP - 2081

EP - 2092

BT - 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018

A2 - Czumaj, Artur

T2 - 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018

Y2 - 7 January 2018 through 10 January 2018

ER -