A PTAS for delay minimization in establishing wireless conference calls

A prevailing feature of mobile telephony systems is that the location of a mobile user may be unknown. Therefore, when the system has to establish a call between users, it may need to search (or page) all the cells that it suspects the users may be located in, in order to find the cells where the users currently reside. The searching process consumes expensive wireless links which motivate search techniques that page as few cells as possible. We consider cellular systems with n cells and m mobile users roaming among the cells. The location of the users is uncertain and is given by m probability distribution vectors. Whenever the system needs to find the users, it conducts a search operation lasting at most d rounds. In each round the system may check an arbitrary subset of cells to see which users are located there. The problem of finding a single user (that is, the case m = 1) is known to be polynomially solvable, whereas the problem of finding any other constant number of users (m ≥ 2) in any fixed (constant) number of rounds (at least two rounds) is known to be NP-hard. In this paper we present a polynomial-time approximation scheme for this problem with a constant number of rounds and a constant number of users. This result improves an earlier frac(e, e - 1) ∼ 1.581977-approximation of Bar-Noy and Malewicz (that applies to any number of users and rounds).