Graphs cannot be indexed in polynomial time for sub-quadratic time string matching, unless SETH fails

Abstract

We consider the following string matching problem on a node-labeled graph $G=(V,E)$: given a pattern string $P$, decide whether there exists a path in $G$ whose concatenation of node labels equals $P$. This is a basic primitive in various problems in bioinformatics, graph databases, or networks. The hardness results of Backurs and Indyk (FOCS 2016) imply that this problem cannot be solved in better than $O(|E||P|)$ time, under the Orthogonal Vectors Hypothesis (OVH), and this holds even under various restrictions on the graph (Equi et al., ICALP 2019). In this paper we consider its offline version, namely the one in which we are allowed to index the graph in order to support time-efficient string matching queries. Indeed, it was tantalizing in the string matching community to believe that sub-quadratic time queries can be achieved, e.g. at the cost of a high-degree polynomial-time indexing. We disprove this belief, showing that, under OVH, no polynomial-time index can support querying $P$ in time $O(|E|^\delta|P|^\beta)$, with either $\delta < 1$ or $\beta < 1$. We prove this tight bound employing a known self-reducibility technique, e.g. from the field of dynamic algorithms, which translates conditional lower bounds for an online problem to its offline version. As a side-contribution, we formalize this technique with the notion of linear independent-components reduction, allowing for a simple proof of our result. As another illustration of our technique, we also translate the quadratic conditional lower bound of Backurs and Indyk (STOC 2015) for the problem of matching a query string inside a text, under edit distance. We obtain an analogous tight quadratic lower bound for its offline version, improving the recent result of Cohen-Addad, Feuilloley and Starikovskaya (SODA 2019), but with a slightly different boundary condition.