Towards Parameterized Hardness on Maintaining Conjunctive Queries

We investigate the fine-grained complexity of dynamically maintaining the result of fixed self-join free conjunctive queries under single-tuple updates. Prior work shows that free-connex queries can be maintained in update time $O(|D|^{\delta})$ for some $\delta \in [0.5, 1]$, where $|D|$ is the size of the current database. However, a gap remains between the best known upper bound of $O(|D|)$ and lower bounds of $\Omega(|D|^{0.5-\epsilon})$ for any $\epsilon \ge 0$. We narrow this gap by introducing two structural parameters to quantify the dynamic complexity of a conjunctive query: the height $k$ and the dimension $d$. We establish new fine-grained lower bounds showing that any algorithm maintaining a query with these parameters must incur update time $\Omega(|D|^{1-1/\max(k,d)-\epsilon})$, unless widely believed conjectures fail. These yield the first super-$\sqrt{|D|}$ lower bounds for maintaining free-connex queries, and suggest the tightness of current algorithms when considering arbitrarily large $k$ and~$d$. Complementing our lower bounds, we identify a data-dependent parameter, the generalized $H$-index $h(D)$, which is upper bounded by $|D|^{1/d}$, and design an efficient algorithm for maintaining star queries, a common class of height 2 free-connex queries. The algorithm achieves an instance-specific update time $O(h(D)^{d-1})$ with linear space $O(|D|)$. This matches our parameterized lower bound and provides instance-specific performance in favorable cases.