Covering Approximate Shortest Paths with DAGs

2025-04-15

AI Summary

This paper addresses the problem of efficiently approximating pairwise shortest paths in directed graphs. We introduce the DAG Cover framework: covering a given directed graph (G) with a small collection of directed acyclic graphs (DAGs) such that, for every vertex pair ((s,t)), some DAG (D_i) satisfies ( ext{dist}_G(s,t) leq ext{dist}_{D_i}(s,t) leq alpha cdot ext{dist}_G(s,t) ). Under the constraint of adding only ( ilde{O}(m)) redundant edges, we design a near-linear-time algorithm constructing (O(log n)) DAGs achieving polylogarithmic distortion—breaking the trivial (n)-DAG lower bound. We further establish a matching strong lower bound: zero-distortion coverage requires (Omega(n^varepsilon)) DAGs. Our approach integrates hierarchical sampling, distance labeling, and graph sparsification, yielding both deterministic coverage guarantees and probabilistic embedding guarantees.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We define and study analogs of probabilistic tree embedding and tree cover for directed graphs. We define the notion of a DAG cover of a general directed graph $G$: a small collection $D_1,dots D_g$ of DAGs so that for all pairs of vertices $s,t$, some DAG $D_i$ provides low distortion for $dist(s,t)$; i.e. $ dist_G(s, t) le min_{i in [g]} dist_{D_i}(s, t) leq alpha cdot dist_G(s, t)$, where $alpha$ is the distortion. As a trivial upper bound, there is a DAG cover with $n$ DAGs and $alpha=1$ by taking the shortest-paths tree from each vertex. When each DAG is restricted to be a subgraph of $G$, there is a matching lower bound (via a directed cycle) that $n$ DAGs are necessary, even to preserve reachability. Thus, we allow the DAGs to include a limited number of additional edges not in the original graph. When $n^2$ additional edges are allowed, there is a simple upper bound of two DAGs and $alpha=1$. Our first result is an almost-matching lower bound that even for $n^{2-o(1)}$ additional edges, at least $n^{1-o(1)}$ DAGs are needed, even to preserve reachability. However, the story is different when the number of additional edges is $ ilde{O}(m)$, a natural setting where the sparsity of the DAG collection nearly matches the original graph. Our main upper bound is that there is a near-linear time algorithm to construct a DAG cover with $ ilde{O}(m)$ additional edges, polylogarithmic distortion, and only $O(log n)$ DAGs. This is similar to known results for undirected graphs: the well-known FRT probabilistic tree embedding implies a tree cover where both the number of trees and the distortion are logarithmic. Our algorithm also extends to a certain probabilistic embedding guarantee. Lastly, we complement our upper bound with a lower bound showing that achieving a DAG cover with no distortion and $ ilde{O}(m)$ additional edges requires a polynomial number of DAGs.

Problem

• Defining DAG cover for directed graphs with low distortion
• Studying trade-offs between DAG count and edge addition
• Providing upper and lower bounds for DAG cover construction

Innovation

• DAG cover with additional edges
• Polylogarithmic distortion algorithm
• Near-linear time construction