Title: Binary Search in Graphs Revisited,
Abstract: In the classical binary search in a path the aim is to detect an unknown
target by asking as few queries as possible, where each query reveals the
direction to the target. This binary search algorithm has been recently
extended by [Emamjomeh-Zadeh et al., STOC, 2016] to the problem of detecting a
target in an arbitrary graph. Similarly to the classical case in the path, the
algorithm of Emamjomeh-Zadeh et al. maintains a candidates' set for the target,
while each query asks an appropriately chosen vertex-- the "median"--which
minimises a potential $\Phi$ among the vertices of the candidates' set. In this
paper we address three open questions posed by Emamjomeh-Zadeh et al., namely
(a) detecting a target when the query response is a direction to an
approximately shortest path to the target, (b) detecting a target when querying
a vertex that is an approximate median of the current candidates' set (instead
of an exact one), and (c) detecting multiple targets, for which to the best of
our knowledge no progress has been made so far. We resolve questions (a) and
(b) by providing appropriate upper and lower bounds, as well as a new potential
$\Gamma$ that guarantees efficient target detection even by querying an
approximate median each time. With respect to (c), we initiate a systematic
study for detecting two targets in graphs and we identify sufficient conditions
on the queries that allow for strong (linear) lower bounds and strong
(polylogarithmic) upper bounds for the number of queries. All of our positive
results can be derived using our new potential $\Gamma$ that allows querying
approximate medians.