Towards Parameterized Hardness on Maintaining Conjunctive Queries

TL;DR

This paper explores the dynamic complexity of maintaining conjunctive queries, introducing structural parameters to establish new lower bounds and designing an efficient algorithm for star queries that matches these bounds.

Contribution

It introduces structural parameters to quantify query complexity, establishes new lower bounds, and provides an instance-specific algorithm for star queries that aligns with these bounds.

Findings

Established super-$ oot{2}{}$ lower bounds for free-connex queries.

Introduced structural parameters: height $k$ and dimension $d$.

Designed an efficient, instance-specific algorithm for star queries.

Abstract

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.