TY - GEN

T1 - Fast Approximate Counting of Cycles

AU - Censor-Hillel, Keren

AU - Even, Tomer

AU - Williams, Virginia Vassilevska

N1 - Publisher Copyright: © Keren Censor-Hillel, Tomer Even, and Virginia Vassilevska Williams.

PY - 2024/7

Y1 - 2024/7

N2 - We consider the problem of approximate counting of triangles and longer fixed length cycles in directed graphs. For triangles, Tětek [ICALP’22] gave an algorithm that returns a (1 ± ε)-approximation in Õ(nω/tω−2) time, where t is the unknown number of triangles in the given n node graph and ω < 2.372 is the matrix multiplication exponent. We obtain an improved algorithm whose running time is, within polylogarithmic factors the same as that for multiplying an n × n/t matrix by an n/t × n matrix. We then extend our framework to obtain the first nontrivial (1 ± ε)-approximation algorithms for the number of h-cycles in a graph, for any constant h ≥ 3. Our running time is Õ(MM (n, n/t1/(h−2), n)), the time to multiply n ×t1/(h−2) byt1/(h−2) × n matrices. n n Finally, we show that under popular fine-grained hypotheses, this running time is optimal.

AB - We consider the problem of approximate counting of triangles and longer fixed length cycles in directed graphs. For triangles, Tětek [ICALP’22] gave an algorithm that returns a (1 ± ε)-approximation in Õ(nω/tω−2) time, where t is the unknown number of triangles in the given n node graph and ω < 2.372 is the matrix multiplication exponent. We obtain an improved algorithm whose running time is, within polylogarithmic factors the same as that for multiplying an n × n/t matrix by an n/t × n matrix. We then extend our framework to obtain the first nontrivial (1 ± ε)-approximation algorithms for the number of h-cycles in a graph, for any constant h ≥ 3. Our running time is Õ(MM (n, n/t1/(h−2), n)), the time to multiply n ×t1/(h−2) byt1/(h−2) × n matrices. n n Finally, we show that under popular fine-grained hypotheses, this running time is optimal.

KW - Approximate cycle counting Fast matrix multiplication

KW - Approximate triangle counting

KW - Fast rectangular matrix multiplication

UR - http://www.scopus.com/inward/record.url?scp=85198390012&partnerID=8YFLogxK

U2 - https://doi.org/10.4230/LIPIcs.ICALP.2024.37

DO - https://doi.org/10.4230/LIPIcs.ICALP.2024.37

M3 - منشور من مؤتمر

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024

A2 - Bringmann, Karl

A2 - Grohe, Martin

A2 - Puppis, Gabriele

A2 - Svensson, Ola

T2 - 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024

Y2 - 8 July 2024 through 12 July 2024

ER -