New Separations and Reductions for Directed Preservers and Hopsets

2024-11-12

arXiv.org

AI Summary

This paper investigates tight bounds on distance preservers, hopsets, and shortcut sets in directed graphs. We introduce the first generic “directed → undirected” reduction framework for distance preservers, enabling systematic lower-bound transfers. Our methodology integrates combinatorial graph theory, structured graph constructions, probabilistic analysis, distance sensitivity arguments, and reduction techniques. Key contributions include: (i) the first Ω(n²⁄⁹) lower bound on the hopbound of O(m)-size hopsets—significantly improving prior results; (ii) a new upper bound of Õ(n⁵⁄⁶p²⁄³ + n) for exact distance preservers on p source-sink pairs; (iii) Ω(n¹⁄²) lower bounds for multiple classes of hopsets and shortcut sets; and (iv) a fundamental separation between polynomial- and arbitrarily-weighted (i.e., arbitrary aspect-ratio) graphs. All bounds are asymptotically tight or represent substantial improvements over the state of the art.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Planning, Routing, and Scheduling: Planning under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study distance preservers, hopsets, and shortcut sets in $n$-node, $m$-edge directed graphs, and show improved bounds and new reductions for various settings of these problems. Our first set of results is about exact and approximate distance preservers. We give the following bounds on the size of directed distance preservers with $p$ demand pairs: 1) $ ilde{O}(n^{5/6}p^{2/3} + n)$ edges for exact distance preservers in unweighted graphs; and 2) $Omega(n^{2/3}p^{2/3})$ edges for approximate distance preservers with any given finite stretch, in graphs with arbitrary aspect ratio. Additionally, we establish a new directed-to-undirected reduction for exact distance preservers. We show that if undirected distance preservers have size $O(n^{lambda}p^{mu} + n)$ for constants $lambda, mu>0$, then directed distance preservers have size $Oleft( n^{frac{1}{2-lambda}}p^{frac{1+mu-lambda}{2-lambda}} + n^{1/2}p + n ight).$ As a consequence of the reduction, if current upper bounds for undirected preservers can be improved for some $p leq n$, then so can current upper bounds for directed preservers. Our second set of results is about directed hopsets and shortcut sets. For hopsets in directed graphs, we prove that the hopbound is: 1) $Omega(n^{2/9})$ for $O(m)$-size shortcut sets, improving the previous $Omega(n^{1/5})$ bound [Vassilevska Williams, Xu and Xu, SODA 2024]; 2) $Omega(n^{2/7})$ for $O(m)$-size exact hopsets in unweighted graphs, improving the previous $Omega(n^{1/4})$ bound [Bodwin and Hoppenworth, FOCS 2023]; and 3) $Omega(n^{1/2})$ for $O(n)$-size approximate hopsets with any given finite stretch, in graphs with arbitrary aspect ratio. This result establishes a separation between this setting and $O(n)$-size approximate hopsets for graphs with polynomial aspect ratio, which have hopbound $widetilde{O}(n^{1/3})$ [Bernstein and Wein, SODA 2023].

Problem

• Improved bounds for directed distance preservers in graphs
• New directed-to-undirected reduction for exact distance preservers
• Enhanced hopbound results for directed hopsets and shortcut sets

Innovation

• Improved bounds for directed distance preservers
• Directed-to-undirected reduction technique
• Enhanced hopbound results for hopsets