TY - JOUR

T1 - Testing Bounded Arboricity

AU - Eden, Talya

AU - Levi, Reut

AU - Ron, Dana

N1 - Publisher Copyright:
© 2020 ACM.

PY - 2020/4

Y1 - 2020/4

N2 - In this article, we consider the problem of testing whether a graph has bounded arboricity. The class of graphs with bounded arboricity includes many important graph families (e.g., planar graphs and randomly generated preferential attachment graphs). Graphs with bounded arboricity have been studied extensively in the past, particularly because for many problems, they allow for much more efficient algorithms and/or better approximation ratios. We present a tolerant tester in the general-graphs model. The general-graphs model allows access to degree and neighbor queries, and the distance is defined with respect to the actual number of edges. Namely, we say that a graph G is ϵ-close to having arboricity α if by removing at most an ϵ-fraction of its edges, we can obtain a graph G′ that has arboricity α, and otherwise we say that G is ϵ-far. Our algorithm distinguishes between graphs that are ϵ-close to having arboricity α and graphs that are c á ϵ-far from having arboricity 3α, where c is an absolute small constant. The query complexity and running time of the algorithm are Õ (n / ϵm) + (1 / ϵ)O(log(1/ϵ)), where n denotes the number of vertices and m denotes the number of edges (we use the notation Õ to hide poly-logarithmic factors in n). In terms of the dependence on n and m, this bound is optimal up to poly-logarithmic factors since ω(n / m) queries are necessary.

AB - In this article, we consider the problem of testing whether a graph has bounded arboricity. The class of graphs with bounded arboricity includes many important graph families (e.g., planar graphs and randomly generated preferential attachment graphs). Graphs with bounded arboricity have been studied extensively in the past, particularly because for many problems, they allow for much more efficient algorithms and/or better approximation ratios. We present a tolerant tester in the general-graphs model. The general-graphs model allows access to degree and neighbor queries, and the distance is defined with respect to the actual number of edges. Namely, we say that a graph G is ϵ-close to having arboricity α if by removing at most an ϵ-fraction of its edges, we can obtain a graph G′ that has arboricity α, and otherwise we say that G is ϵ-far. Our algorithm distinguishes between graphs that are ϵ-close to having arboricity α and graphs that are c á ϵ-far from having arboricity 3α, where c is an absolute small constant. The query complexity and running time of the algorithm are Õ (n / ϵm) + (1 / ϵ)O(log(1/ϵ)), where n denotes the number of vertices and m denotes the number of edges (we use the notation Õ to hide poly-logarithmic factors in n). In terms of the dependence on n and m, this bound is optimal up to poly-logarithmic factors since ω(n / m) queries are necessary.

KW - Property testing

KW - graph arboricity

KW - graph degeneracy

KW - tolerant testing

UR - http://www.scopus.com/inward/record.url?scp=85084762512&partnerID=8YFLogxK

U2 - 10.1145/3381418

DO - 10.1145/3381418

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AN - SCOPUS:85084762512

SN - 1549-6325

VL - 16

JO - ACM Transactions on Algorithms

JF - ACM Transactions on Algorithms

IS - 2

M1 - 18

ER -