On the Hardness of Wildcard Pattern Matching on de Bruijn Graphs

Abstract

In the pattern matching on labeled graphs problem, given an edge labeled graph $G = (V, E)$ and a string P, one seeks to identify if there exists a walk in the graph whose concatenation of edge labels (approximately) matches P. This is an elementary subproblem for utilizing genome graphs to represent collections of genetic sequences where patterns arise as reads in the sequencing data. Unfortunately, for general graphs, it is known that an algorithm running in $O(|E||P|^{1-\varepsilon } + |E|^{1-\varepsilon }|P|)$ time for constant $\varepsilon > 0$ is not possible under the Strong Exponential Time Hypothesis (SETH). De Bruijn graphs provide a valuable exception, allowing for a path exactly matching a pattern to be found in $O(|E| + |P|)$ for constant-sized alphabets. This property has led de Bruijn graphs to be applied as indexes in the popular tool vg-toolkit. In this work, we consider the case where wildcards (that match with any edge label) are included in the pattern, and the graph is a de Bruijn graph. We demonstrate that adding these wildcards to the pattern is enough to again prove quadratic lower bounds conditioned on SETH for pattern matching on de Bruijn graphs, even when restricted to alphabets of size at most three and k-mer length $\varTheta (\log |V|)$.