Approximation Algorithms for Optimal Hopsets
Michael Dinitz, Ama Koranteng, Yasamin Nazari
TL;DR
This work initiates and develops the optimization variant of hopsets: given a graph $G$, the goal is to add the smallest possible set of edges to achieve a hopbound $\beta$ and stretch $\alpha$ for all demand pairs. The authors propose three main algorithmic paradigms—LP relaxation with a (1+ε)-approximate solver, star sampling with randomized LP rounding, and a layered-junction-tree construction—and show how they interact with existing existential hopset bounds to yield regime-dependent guarantees for both directed and undirected graphs. A key technical achievement is a PTAS for the Hopbounded Restricted Shortest Path problem used as a separation oracle, plus a layered-graph reduction that enables junction-tree techniques in the hopset setting. In addition to delivering substantial upper bounds, the paper proves strong hardness results for directed hopsets via a Min-Rep/Label Cover reduction, establishing subpolynomial inapproximability under standard complexity assumptions. Collectively, these results bridge optimization and existential perspectives on hopsets, offering practical, input-specific hopset constructions with provable guarantees across diverse regimes and problem parameters.
Abstract
For a given graph $G$, a "hopset" $H$ with hopbound $β$ and stretch $α$ is a set of edges such that between every pair of vertices $u$ and $v$, there is a path with at most $β$ hops in $G \cup H$ that approximates the distance between $u$ and $v$ up to a multiplicative stretch of $α$. 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 $3$.