Lower bounds on the competitive ratio for mobile user tracking and distributed job scheduling

The authors prove a lower bound of Omega (log n/log log n) on the competitive ratio of any (deterministic or randomised) distributed algorithm for solving the mobile user problem on certain networks of n processors. The lower bound holds for various networks, including the hypercube, any network with sufficiently large girth, and any highly expanding graph. A similar Omega (log n/log log n) lower bound is proved for the competitive ratio of the maximum job delay of any distributed algorithm for solving a distributed scheduling problem on any of these networks. The proofs combine combinatorial techniques with tools from linear algebra and harmonic analysis and apply, in particular, a generalization of the vertex isoperimetric problem on the hypercube, which may be of independent interest.