Distributed Complexity of $P_k$-freeness: Decision and Certification

The class of graphs that do not contain a path on $k$ nodes as an induced subgraph ($P_k$-free graphs) has rich applications in the theory of graph algorithms. This paper explores the problem of deciding $P_k$-freeness from the viewpoint of distributed computing. For specific small values of $k$, we present the \textit{first} $\mathsf{CONGEST}$ algorithms specified for $P_k$-freeness, utilizing structural properties of $P_k$-free graphs in a novel way. Specifically, we show that $P_k$-freeness can be decided in $\tilde{O}(1)$ rounds for $k=4$ in the $\mathsf{broadcast\;CONGEST}$ model, and in $\tilde{O}(n)$ rounds for $k=5$ in the $\mathsf{CONGEST}$ model, where $n$ is the number of nodes in the network and $\tilde{O}(\cdot)$ hides a $\mathrm{polylog}(n)$ factor. These results significantly improve the previous $O(n^{2-2/(3k+2)})$ upper bounds by Eden et al. (Dist.~Comp.~2022). We also construct a local certification of $P_5$-freeness with certificates of size $\tilde{O}(n)$. This is nearly optimal, given our $\Omega(n^{1-o(1)})$ lower bound on certificate size, and marks a significant advancement as no nontrivial bounds for proof-labeling schemes of $P_5$-freeness were previously known. For general $k$, we establish the first $\mathsf{CONGEST}$ lower bound, which is of the form $n^{2-1/\Theta(k)}$. The $n^{1/\Theta(k)}$ factor is unavoidable, in view of the $O(n^{2-2/(3k+2)})$ upper bound mentioned above. Additionally, our approach yields the \textit{first} superlinear lower bound on certificate size for local certification. This partially answers the conjecture on the optimal certificate size of $P_k$-freeness, asked by Bousquet et al. (arXiv:2402.12148). Finally, we propose a novel variant of the problem called ordered $P_k$ detection, and show a linear lower bound and its nontrivial connection to distributed subgraph detection.