Affiliation:
1. ETH Zürich, Zürich, Switzerland
2. School of Mathematics, Tel Aviv University, Tel Aviv, Israel
3. Department of Mathematics, University of Haifa, Haifa, Israel
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
C
2k
and
C
2k+1
in
n
-vertex graphs of bounded degeneracy in time Õ(
n
d
k
), where the fastest
known
algorithm for detecting directed copies of
C
k
in general
m
-edge digraphs runs in time Õ(
m
d
k
).
Conversely, one can transform any
O(n
b
k
)
algorithm for computing the number of homomorphic copies of
C
2k
or of
C
2k+1
in
n
-vertex graphs of bounded degeneracy, into an Õ(
m
b
k
) time algorithm for detecting directed copies of
C
k
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
C
k
-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.
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
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