Listing, Verifying and Counting Lowest Common Ancestors in DAGs: Algorithms and Fine-Grained Lower Bounds

The AP-LCA problem asks, given an $n$-node directed acyclic graph (DAG), to compute for every pair of vertices $u$ and $v$ in the DAG a lowest common ancestor (LCA) of $u$ and $v$ if one exists. In this paper we study several interesting variants of AP-LCA, providing both algorithms and fine-grained lower bounds for them. The lower bounds we obtain are the first conditional lower bounds for LCA problems higher than $n^{\omega -o(1)}$, where $\omega$ is the matrix multiplication exponent. Some of our results include:

• In any DAG, we can detect all vertex pairs that have at most two LCAs and list all of their LCAs in $O(n^{\omega })$ time. This algorithm extends a result of [Kowaluk and Lingas ESA’07] which showed an $\tilde{O}(n^{\omega })$ time algorithm that detects all pairs with a unique LCA in a DAG and outputs their corresponding LCAs.
• Listing $7$ LCAs per vertex pair in DAGs requires $n^{3-o(1)}$ time under the popular assumption that 3-uniform 5-hyperclique detection requires $n^{5-o(1)}$ time. This is surprising since essentially cubic time is sufficient to list all LCAs (if $\omega =2$).
• Counting the number of LCAs for every vertex pair in a DAG requires $n^{3-o(1)}$ time under the Strong Exponential Time Hypothesis, and $n^{\omega (1,2,1)-o(1)}$ time under the $4$-Clique hypothesis. This shows that the algorithm of [Echkardt, M"{u}hling and Nowak ESA’07] for listing all LCAs for every pair of vertices is likely optimal.
• Given a DAG and a vertex $w_{u,v}$ for every vertex pair $u,v$, verifying whether all $w_{u,v}$ are valid LCAs requires $n^{2.5-o(1)}$ time assuming 3-uniform 4-hyperclique requires $n^{4-o(1)}$ time. This defies the common intuition that verification is easier than computation since returning some LCA per vertex pair can be solved in $O(n^{2.447})$ time [Grandoni et al. SODA’21].