Compact Path Representation in DAGs via Colored Edge Pebbling

2026-08-13

AI Summary

This study addresses the problem of compactly representing multiple paths in a directed acyclic graph (DAG) to support efficient querying. The authors propose a novel framework based on colored-edge pebbling, wherein “pebbles” of distinct colors are placed on edges such that each predefined path can be uniquely reconstructed. Their key contributions include formulating the minimal pebbling problem as one solvable in polynomial time, establishing a theoretical connection between saturated pebbling and the set cover problem—and proving its NP-hardness—and integrating graph-theoretic modeling, integer linear programming, and minimum-weight set cover algorithms to achieve a compact representation whose space complexity depends only on the number of pebbles. Experimental results demonstrate that the proposed saturated pebbling strategy significantly improves the efficiency of both path and edge queries.

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.

Problem

• variation graph
• compact representation
• path reconstruction
• colored edge pebbling
• directed acyclic graph

Innovation

• colored edge pebbling
• variation graph
• directed acyclic graph
• minimum-weight set cover
• integer linear programming