Multi-marked analogue of the averaged-walk tradeoff

Determine whether a useful analogue of the averaged-walk hitting-time guarantee exists for a marked set containing multiple states, or construct a counterexample showing that no such analogue holds at this level of generality.

Background

The main theorem establishes the simultaneous update/checking-cost tradeoff only for a single marked state. Its proof uses the fact that, after the original walk first reaches the unique marked state, the state is known exactly, allowing the remaining expected search time to be related to a return time through Kac’s lemma.

For a marked set with several states, the first state hit within the set may have a distribution biased by the dynamics of the chain, and this distribution is not determined solely by the stationary marked mass. The authors therefore explicitly leave open whether a corresponding multi-marked result can be proved or whether a counterexample exists.