Codes, Lower Bounds, and Phase Transitions in the Symmetric Rendezvous Problem


Tom Hayes


University of New Mexico
Department of Computer Science
Albuquerque, NM 87131
United States of America


Tuesday, 15 November 2016, 16:00 to 17:00



In the rendezvous problem, two parties with different labelings of the vertices of a complete graph are trying to meet at some vertex at the same time. It is well-known that if the parties have predetermined roles, then the strategy where one of them waits at one vertex, while the other visits all n vertices in random order is optimal, taking at most n steps and averaging about inline image. Anderson and Weber (J. Appl. Prob. 1990, pp. 839–851) considered the symmetric rendezvous problem, where both parties must use the same randomized strategy. They analyzed strategies where the parties repeatedly play the optimal asymmetric strategy, determining their role independently each time by a biased coin-flip. By tuning the bias, Anderson and Weber achieved an expected meeting time of about inline image, which they conjectured to be asymptotically optimal.

We change perspective slightly: instead of minimizing the expected meeting time, we seek to maximize the probability of meeting within a specified time T. The Anderson-Weber strategy, which fails with constant probability when inline image, is not asymptotically optimal for large T in this setting. Specifically, we exhibit a symmetric strategy that succeeds with probability inline image in inline image steps. This is tight: for any inline image, any symmetric strategy with inline image fails with constant probability. Our strategy uses a new combinatorial object that we dub a “rendezvous code,” which may be of independent interest.

When inline image, we show that the probability of meeting within T steps is indeed asymptotically maximized by the Anderson-Weber strategy. Our results imply new lower bounds, showing that the best symmetric strategy takes at least inline image steps in expectation. We also present some partial results for the symmetric rendezvous problem on other vertex-transitive graphs. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 742–765, 2016.