Counting Homomorphic Cycles in Degenerate Graphs

Since computing most variants of the subgraph counting problem in general graphs is conjectured to be hard, 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 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: 1. One can compute the number of homomorphic copies of $C_{2k}$ and $C_{2k+1}$ in $n$-vertex graphs of bounded degeneracy in time $O(n^{d_{k}})$, where $d_k$ is the exponent of the fastest known algorithm for detecting directed copies of $C_k$ in general $m$-edge digraphs. 2. Conversely, one can transform any $n^{b_{k}}$ 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 $\tilde{O}(m^{b_{k}})$ 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 $C_{2k}$-homomorphisms (or $C_{2k+1}$-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 $k$-cycles in general digraphs, which was carried out in a recent breakthrough of Dalirrooyfard, Vuong and Vassilevska Williams.