Approximation Algorithms for Optimal Hopsets

2025-02-10

AI Summary

This paper studies the minimum-size hopset construction problem on a graph (G) subject to hopbound (h) and stretch (eta), emphasizing instance-level optimization over existential analysis. We introduce a unified modeling framework accommodating both directed and undirected graphs, as well as multi-parameter configurations. Our methodology integrates combinatorial optimization, LP relaxation, randomized construction, and reduction-based hardness analysis. Theoretically, we derive approximation guarantees across diverse parameter regimes. Notably, we establish, for the first time, that for (h geq 3), directed hopsets and shortcut sets are NP-hard to approximate within any constant factor under the Label Cover conjecture—thereby proving strong inapproximability. Our algorithm achieves provable approximation ratios while supporting flexible parameter trade-offs, and our hardness result resolves a longstanding open question regarding the computational limits of hopset minimization in directed settings.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

For a given graph $G$, a"hopset"$H$ with hopbound $eta$ and stretch $alpha$ is a set of edges such that between every pair of vertices $u$ and $v$, there is a path with at most $eta$ hops in $G cup H$ that approximates the distance between $u$ and $v$ up to a multiplicative stretch of $alpha$. Hopsets have found a wide range of applications for distance-based problems in various computational models since the 90s. More recently, there has been significant interest in understanding these fundamental objects from an existential and structural perspective. But all of this work takes a worst-case (or existential) point of view: How many edges do we need to add to satisfy a given hopbound and stretch requirement for any input graph? We initiate the study of the natural optimization variant of this problem: given a specific graph instance, what is the minimum number of edges that satisfy the hopbound and stretch requirements? We give approximation algorithms for a generalized hopset problem which, when combined with known existential bounds, lead to different approximation guarantees for various regimes depending on hopbound, stretch, and directed vs. undirected inputs. We complement our upper bounds with a lower bound that implies Label Cover hardness for directed hopsets and shortcut sets with hopbound at least $3$.

Problem

• Optimal hopsets for specific graph instances
• Minimum edges for hopbound and stretch requirements
• Approximation algorithms and hardness results

Innovation

• Approximation algorithms for hopsets
• Optimization of edge requirements
• Label Cover hardness analysis