Faster Symmetric Rendezvous on Four or More Locations

2026-04-02

AI Summary

This study addresses the symmetric rendezvous problem, where two players must adopt identical strategies to visit one of $n$ locations with the goal of minimizing the expected time until their first meeting. While the classic Anderson–Weber strategy has long been a focal point for $n \geq 4$, its optimality in this regime remains unresolved. By integrating probabilistic strategy design, stochastic process analysis, and combinatorial optimization, this work rigorously establishes—for the first time—that the Anderson–Weber strategy is suboptimal for all $n \geq 4$. Moreover, the authors construct novel symmetric strategies that achieve strictly lower expected rendezvous times. These results transcend the conventional round-based structural constraints inherent in prior approaches, thereby advancing the theoretical foundations of symmetric rendezvous search.

Abstract

In the symmetric rendezvous problem two players follow the same (randomized) strategy to visit one of $n$ locations in each time step $t=0,1,2,\dots$. Their goal is to minimize the expected time until they visit the same location and thus meet. Anderson and Weber [J. Appl. Prob., 1990] proposed a strategy that operates in rounds of $n-1$ steps: a player either remains in one location for $n-1$ steps or visits the other $n-1$ locations in random order; the choice between these two options is made with a probability that depends only on $n$. The strategy is known to be optimal for $n=2$ and $n=3$, and there is convincing evidence that it is not optimal for $n=4$. We show that it is not optimal for any $n\geq 4$, by constructing a strategy with a smaller expected meeting time.

Problem

• symmetric rendezvous
• expected meeting time
• optimal strategy
• randomized strategy
• location visitation

Innovation

• symmetric rendezvous
• expected meeting time
• randomized strategy
• optimality
• multi-location search