AI 中文总结
本文提出以拓扑为中心的彩色边 pebbling 框架，解决变异图紧凑表示问题，最小规模 pebbling 可多项式求解，饱和 pebbling 为 NP 难但可归约为集合覆盖，能实现更快查询。
AI 中文摘要
紧凑表示变异图是计算泛基因组学的核心问题，通常采用源自文本并适配图的技术来解决。本文提出一种以拓扑为中心视角的新框架：将变异图建模为有向无环图（DAG），并结合一组带不同颜色的指定路径；我们的紧凑表示以图的 pebbling 为核心，即向边放置彩色 pebble，使得每条预定义路径都能从 pebbling 后的边唯一重构。特别地，饱和 pebbling 会为每条选定边标记所有经过该边的路径（颜色）。我们首先提出一种数据结构来表示和查询变异图，其存储空间取决于 pebbling 的规模，支持的查询包括：（i）路径查询，给定颜色即可恢复对应路径；（ii）边查询，报告经过给定边的路径颜色。随后我们证明，寻找最小规模 pebbling 的问题可在多项式时间内求解；相反，寻找最小规模饱和 pebbling 的问题是 NP 难的，但可归约为最小权重集合覆盖问题，使我们能利用整数线性规划（ILP）求解器。我们展示了如何利用饱和 pebbling 实现比最小规模 pebbling 更快的查询时间。该框架为开发更高效的变异图表示提供了新的算法视角，其基础是对这些图拓扑结构的研究。
英文摘要
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.
