TY - GEN
T1 - Counting Homomorphic Cycles in Degenerate Graphs
AU - Gishboliner, Lior
AU - Levanzov, Yevgeny
AU - Shapira, Asaf
AU - Yuster, Raphael
N1 - Publisher Copyright:
© 2022 Association for Computing Machinery. All rights reserved.
PY - 2022
Y1 - 2022
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85130735991&partnerID=8YFLogxK
M3 - פרסום בספר כנס
AN - SCOPUS:85130735991
T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
SP - 417
EP - 430
BT - ACM-SIAM Symposium on Discrete Algorithms, SODA 2022
PB - Association for Computing Machinery
Y2 - 9 January 2022 through 12 January 2022
ER -