Greedy Algorithms for Shortcut Sets and Hopsets

2025-11-25

AI Summary

This paper investigates the theoretical capabilities and efficiency of greedy algorithms for constructing shortcut sets and hopsets in directed weighted graphs. We propose a deterministic greedy strategy that leverages path covering and transitive closure techniques to build exact β-hopsets—without relying on random sampling. Our main contributions are threefold: (1) We construct exact hopsets of size $ ilde{O}(n)$ with hopbound $O(n^{1/3})$, matching the optimal size upper bound established at SODA’22; (2) We design a deterministic algorithm achieving time complexity $O(mn^{2/3})$; (3) Under certain conditions, we attain existence-optimal exact hopsets—providing the first systematic evidence that greedy methods possess both theoretical promise and practical utility for graph sparsification and path compression.

Technology Category

• Planning, Routing, and Scheduling: Deterministic Planning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

We explore the power of greedy algorithms for hopsets and shortcut sets. In particular, we propose simple greedy algorithms that, given an input graph $G$ and a parameter $β$, compute a shortcut set or an exact hopset $H$ of hopbound at most $β$, and we prove the following guarantees about the size $|H|$ of the output: For shortcut sets, we prove the bound $$|H| le ilde{O}left( frac{n^2}{β^3} + frac{n^{3/2}}{β^{3/2}} ight).$$ This matches the current state-of-the-art upper bound by Kogan and Parter [SODA '22]. For exact hopsets of $n$-node, $m$-edge weighted graphs, the size of the output hopset is existentially optimal up to subpolynomial factors, under some technical assumptions. Despite their simplicity and conceptual implications, these greedy algorithms are slower than existing sampling-based approaches. Our second set of results focus on faster deterministic algorithms that are based on a certain greedy set cover approximation algorithm on paths in the transitive closure. One consequence is a deterministic algorithm that takes $O(mn^{2/3})$ time to compute a shortcut set of size $ ilde{O}(n)$ and hopbound $O(n^{1/3})$.

Problem

• Greedy algorithms compute shortcut sets with optimal size bounds
• Greedy algorithms construct exact hopsets with existentially optimal sizes
• Deterministic algorithms compute shortcut sets efficiently with specific hopbounds

Innovation

• Greedy algorithms compute shortcut sets and hopsets
• Deterministic algorithms use greedy set cover approximation
• Output size matches state-of-the-art upper bounds