A Symbolic Programming Approach to the Rendezvous Search Problem

Abstract

In this paper we solve the rendezvous problem on the line with markers that can be dropped at chosen times when the initial distance D between the players is known. In the case of one marker, the $$M_1$$ M 1 game, the marker is held by player II at the start of the game and, once dropped and found by player I, indicates in which direction player I must move. In the case of two markers, the $$M_2$$ M 2 game, each player holds one and the dropping times may differ. There is uncertainty regarding the problem initial configuration, and the goal is to minimize the expected rendezvous time that we call the rendezvous value (of the game) denoted $$R_1$$ R 1 and $$R_2$$ R 2 for the $$M_1$$ M 1 and $$M_2$$ M 2 games respectively. We present an algorithm that computes exactly the rendezvous value of the $$M_1$$ M 1 game as a function of the dropping time z, i.e. $$z\mapsto R_1(z)$$ z ↦ R 1 ( z ) . Then we show that the function $$R_1(z)$$ R 1 ( z ) is locally an affine function and we compute the parameters of the local representations of $$R_1(z)$$ R 1 ( z ) . Finally, the rendezvous value of the game $$R_1=min_z R_1(z)$$ R 1 = m i n z R 1 ( z ) and the optimal dropping times can be determined with the expression of $$R_1(z)$$ R 1 ( z ) . The same proceeding can be extended to apply to the problem $$M_2$$ M 2 . Symbolic execution of programs is a classical technique of program testing in computer science, see King [1] for the pioneering work. In this work we adapt the symbolic execution technique to solve an optimization problem. To our knowledge this is the first time that this is attempted, in particular to deal with rendezvous problems.