{"title":"Counting Homomorphic Cycles in Degenerate Graphs","authors":"Lior Gishboliner, Yevgeny Levanzov, Asaf Shapira, Raphael Yuster","doi":"https://dl.acm.org/doi/10.1145/3560820","DOIUrl":null,"url":null,"abstract":"<p>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.</p><p>Our main result in this paper is a surprisingly tight relation between the above task and the well-studied problem of <i>detecting (standard) copies</i> of directed cycles in <i>general directed</i> graphs. More precisely, we prove the following:\n<p><ul><li><p>One can compute the number of homomorphic copies of <i>C<sub>2k</sub></i> and <i>C<sub>2k+1</sub></i> in <i>n</i>-vertex graphs of bounded degeneracy in time Õ(<i>n<sup>d<sub>k</sub></sup></i>), where the fastest <i>known</i> algorithm for detecting directed copies of <i>C<sub>k</sub></i> in general <i>m</i>-edge digraphs runs in time Õ(<i>m<sup>d<sub>k</sub></sup></i>).</p></li><li><p>Conversely, one can transform any <i>O(n<sup>b<sub>k</sub></sup>)</i> algorithm for computing the number of homomorphic copies of <i>C<sub>2k</sub></i> or of <i>C<sub>2k+1</sub></i> in <i>n</i>-vertex graphs of bounded degeneracy, into an Õ(<i>m<sup>b<sub>k</sub></sup></i>) time algorithm for detecting directed copies of <i>C<sub>k</sub></i> in general <i>m</i>-edge digraphs.</p></li></ul></p></p><p>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 <i>C<sub>k</sub></i>-homomorphisms in degenerate graphs and show that one part of its <i>analysis</i> 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 <i>k</i>-cycles in directed and undirected graphs of bounded degeneracy that is faster than all previously known algorithms for 7 ≤ <i>k</i> ≤ 11, and faster for all <i>k</i> ≥ 7 if the matrix multiplication exponent is 2.</p>","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"1 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2023-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACM Transactions on Algorithms","FirstCategoryId":"94","ListUrlMain":"https://doi.org/https://dl.acm.org/doi/10.1145/3560820","RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
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 Õ(ndk), where the fastest known algorithm for detecting directed copies of Ck in general m-edge digraphs runs in time Õ(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 Õ(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.
期刊介绍：
ACM Transactions on Algorithms welcomes submissions of original research of the highest quality dealing with algorithms that are inherently discrete and finite, and having mathematical content in a natural way, either in the objective or in the analysis. Most welcome are new algorithms and data structures, new and improved analyses, and complexity results. Specific areas of computation covered by the journal include
combinatorial searches and objects;
counting;
discrete optimization and approximation;
randomization and quantum computation;
parallel and distributed computation;
algorithms for
graphs,
geometry,
arithmetic,
number theory,
strings;
on-line analysis;
cryptography;
coding;
data compression;
learning algorithms;
methods of algorithmic analysis;
discrete algorithms for application areas such as
biology,
economics,
game theory,
communication,
computer systems and architecture,
hardware design,
scientific computing
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