Polynomial deterministic rendezvous in arbitrary graphs

The rendezvous problem in graphs has been extensively studied in the literature, mainly using a randomized approach. Two mobile agents have to meet at some node of a connected graph. We study deterministic algorithms for this problem, assuming that agents have distinct identifiers and are located in nodes of an unknown anonymous connected graph. Startup times of the agents are arbitrarily decided by the adversary. The measure of performance of a rendezvous algorithm is its cost: for a given initial location of agents in a graph, this is the number of steps since the startup of the later agent until rendezvous is achieved. Deterministic rendezvous has been previously shown feasible in arbitrary graphs [16] but the proposed algorithm had cost exponential in the number n of nodes and in the smaller identifier l, and polynomial in the difference τ between startup times. The following problem was stated in [16]: Does there exist a deterministic rendezvous algorithm with cost polynomial in n, τ and in labels L₁, L₂ of the agents (or even polynomial in n, τ and log L₁, logL₂)? We give a positive answer to both problems: our main result is a deterministic rendezvous algorithm with cost polynomial in n, τ and log l. We also show a lower bound Ω(n²) on the cost of rendezvous in some family of graphs.