Compact Path Representation in DAGs via Colored Edge Pebbling

Abstract

Compactly representing a variation graph is a core problem in computational pangenomics that is usually attacked with techniques that have been originated on texts and adapted to graphs. In this paper we propose a new framework that takes a topology-centric perspective instead. A variation graph is modeled as a directed acyclic graph (DAG) together with a set of distinguished paths, where each path is assigned a distinct color. Our compact representation is centered on pebbling the graph, i.e. placing colored pebbles on edges so that every predefined path can be univocally reconstructed from the pebbled edges. In particular, a saturated pebbling marks each chosen edge with every path (color) traversing it. We first propose a data structure to represent and query a variation graph with storage space depending on the size of the pebbling. The supported queries are: (i) path query, which recovers a path given its color, and (ii) edge query, which reports the colors of paths traversing a given edge. We then prove that the problem of finding a pebbling of minimum size is solvable in polynomial time. On the contrary, we prove that finding a saturated pebbling of minimum size is NP-hard, but can be reduced to the minimum-weight set cover problem, allowing us to leverage integer linear programming (ILP) solvers. We show how to exploit saturated pebblings to achieve faster queries times than minimum size pebbling. Our framework opens a new algorithmic viewpoint on developing more efficient variation graph representations rooted on the study of the topology of those graphs.