Graphs Cannot Be Indexed in Polynomial Time for Sub-quadratic Time String Matching, Unless SETH Fails

Abstract

The string matching problem on a node-labeled graph G= (V, E) asks whether a given pattern string P has an occurrence in G, in the form of a path whose concatenation of node labels equals P. This is a basic primitive in various problems in bioinformatics, graph databases, or networks, but only recently proven to have a O(|E||P|)-time lower bound, under the Orthogonal Vectors Hypothesis (OVH). We consider here its indexed version, in which we can index the graph in order to support time-efficient string queries. We show that, under OVH, no polynomial-time indexing scheme of the graph can support querying P in time O(| P| + | E| δ| P| β), with either δ< 1 or β< 1. As a side-contribution, we introduce the notion of linear independent-components (lic) reduction, allowing for a simple proof of our result. As another illustration that hardness of indexing follows as a corollary of a lic reduction, 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 indexed version, improving the recent result of Cohen-Addad, Feuilloley and Starikovskaya (SODA 2019), but with a slightly different boundary condition.