Algorithms and Complexity on Indexing Founder Graphs

We study the problem of matching a string in a labeled graph. Previous research has shown that unless the Orthogonal Vectors Hypothesis (OVH) is false, one cannot solve this problem in strongly sub-quadratic time, nor index the graph in polynomial time to answer queries efficiently (Equi et al. ICALP 2019, SOFSEM 2021). These conditional lower-bounds cover even deterministic graphs with binary alphabet, but there naturally exist also graph classes that are easy to index: For example, Wheeler graphs (Gagie et al. Theor. Comp. Sci. 2017) cover graphs admitting a Burrows-Wheeler transform -based indexing scheme. However, it is NP-complete to recognize if a graph is a Wheeler graph (Gibney, Thankachan, ESA 2019). We propose an approach to alleviate the construction bottleneck of Wheeler graphs. Rather than starting from an arbitrary graph, we study graphs induced from multiple sequence alignments (MSAs). Elastic degenerate strings (Bernadini et al. SPIRE 2017, ICALP 2019) can be seen as such graphs, and we introduce here their generalization: elastic founder graphs. We first prove that even such induced graphs are hard to index under OVH. Then we introduce two subclasses, repeat-free and semi-repeat-free graphs, that are easy to index. We give a linear time algorithm to construct a repeat-free (non-elastic) founder graph from a gapless MSA, and (parameterized) near-linear time algorithms to construct a semi-repeat-free (repeat-free, respectively) elastic founder graph from general MSA. Finally, we show that repeat-free founder graphs admit a reduction to Wheeler graphs in polynomial time.

Automated summary

This paper investigates the problem of matching a string in a labeled graph. Previous research has shown that it is difficult to solve this problem in strongly sub-quadratic time unless the Orthogonal Vectors Hypothesis is false. However, there are graph classes, such as Wheeler graphs, that can be easily indexed. This paper proposes a method to alleviate the construction bottleneck of Wheeler graphs by studying graphs induced from multiple sequence alignments. The paper introduces the concept of elastic founder graphs and proves that even such induced graphs are hard to index. It also presents two subclasses, repeat-free and semi-repeat-free graphs, that are easy to index.