Counting Homomorphic Cycles in Degenerate Graphs

Lior Gishboliner, Yevgeny Levanzov, Asaf Shapira, Raphael Yuster

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Since counting subgraphs in general graphs is, by and large, a computationally demanding problem, it is natural to try and design fast algorithms for restricted families of graphs. One such family that has been extensively studied is that of graphs of bounded degeneracy (e.g., planar graphs). This line of work, which started in the early 80's, culminated in a recent work of Gishboliner et al., which highlighted the importance of the task of counting homomorphic copies of cycles (i.e., cyclic walks) in graphs of bounded degeneracy. Our main result in this paper is a surprisingly tight relation between the above task and the well-studied problem of detecting (standard) copies of directed cycles in general directed graphs. More precisely, we prove the following: One can compute the number of homomorphic copies of C2k and C2k+1 in n-vertex graphs of bounded degeneracy in time ~O (ndk ), where the fastest known algorithm for detecting directed copies of Ck in general m-edge digraphs runs in time ~O(mdk ). Conversely, one can transform any O(nbk ) algorithm for computing the number of homomorphic copies of C2k or of C2k+1 in n-vertex graphs of bounded degeneracy, into an ~O(mbk ) time algorithm for detecting directed copies of Ck in general m-edge digraphs. We emphasize that our first result does not use a black-box reduction (as opposed to the second result which does). Instead, we design an algorithm for computing the number of Ck-homomorphisms in degenerate graphs and show that one part of its analysis can be reduced to the analysis of the fastest known algorithm for detecting directed cycles in general digraphs, which was carried out in a recent breakthrough of Dalirrooyfard, Vuong and Vassilevska Williams. As a by-product of our algorithm, we obtain a new algorithm for detecting k-cycles in directed and undirected graphs of bounded degeneracy that is faster than all previously known algorithms for 7 k 11, and faster for all k 7 if the matrix multiplication exponent is 2.

Original languageEnglish
Title of host publicationACM-SIAM Symposium on Discrete Algorithms, SODA 2022
PublisherAssociation for Computing Machinery
Pages417-430
Number of pages14
ISBN (Electronic)9781611977073
StatePublished - 2022
Event33rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2022 - Alexander, United States
Duration: 9 Jan 202212 Jan 2022

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
Volume2022-January

Conference

Conference33rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2022
Country/TerritoryUnited States
CityAlexander
Period9/01/2212/01/22

Fingerprint

Dive into the research topics of 'Counting Homomorphic Cycles in Degenerate Graphs'. Together they form a unique fingerprint.

Cite this