TY - GEN

T1 - Towards a decomposition-optimal algorithm for counting and sampling arbitrary motifs in sublinear time

AU - Biswas, Amartya Shankha

AU - Eden, Talya

AU - Rubinfeld, Ronitt

N1 - Publisher Copyright:
© Amartya Shankha Biswas, Talya Eden, and Ronitt Rubinfeld; licensed under Creative Commons License CC-BY 4.0

PY - 2021/9/1

Y1 - 2021/9/1

N2 - We consider the problem of sampling and approximately counting an arbitrary given motif H in a graph G, where access to G is given via queries: degree, neighbor, and pair, as well as uniform edge sample queries. Previous algorithms for these tasks were based on a decomposition of H into a collection of odd cycles and stars, denoted D∗(H) = {Ok1,..., Okq, Sp1,..., Spℓ}. These algorithms were shown to be optimal for the case where H is a clique or an odd-length cycle, but no other lower bounds were known. We present a new algorithm for sampling arbitrary motifs which, up to poly(log n) factors, is always at least as good, and for most graphs G is strictly better. The main ingredient leading to this improvement is an improved uniform algorithm for sampling stars, which might be of independent interest, as it allows to sample vertices according to the p-th moment of the degree distribution. Finally, we prove that this algorithm is decomposition-optimal for decompositions that contain at least one odd cycle. These are the first lower bounds for motifs H with a nontrivial decomposition, i.e., motifs that have more than a single component in their decomposition.

AB - We consider the problem of sampling and approximately counting an arbitrary given motif H in a graph G, where access to G is given via queries: degree, neighbor, and pair, as well as uniform edge sample queries. Previous algorithms for these tasks were based on a decomposition of H into a collection of odd cycles and stars, denoted D∗(H) = {Ok1,..., Okq, Sp1,..., Spℓ}. These algorithms were shown to be optimal for the case where H is a clique or an odd-length cycle, but no other lower bounds were known. We present a new algorithm for sampling arbitrary motifs which, up to poly(log n) factors, is always at least as good, and for most graphs G is strictly better. The main ingredient leading to this improvement is an improved uniform algorithm for sampling stars, which might be of independent interest, as it allows to sample vertices according to the p-th moment of the degree distribution. Finally, we prove that this algorithm is decomposition-optimal for decompositions that contain at least one odd cycle. These are the first lower bounds for motifs H with a nontrivial decomposition, i.e., motifs that have more than a single component in their decomposition.

KW - Approximate counting

KW - Graph algorithms

KW - Sampling subgraphs

KW - Sublinear time algorithms

UR - http://www.scopus.com/inward/record.url?scp=85112362551&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs-APPROX/RANDOM.2021.55

DO - 10.4230/LIPIcs-APPROX/RANDOM.2021.55

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AN - SCOPUS:85112362551

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2021

A2 - Wootters, Mary

A2 - Sanita, Laura

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 24th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2021 and 25th International Conference on Randomization and Computation, RANDOM 2021

Y2 - 16 August 2021 through 18 August 2021

ER -