  We consider a classic search problem first proposed by S. Gal in which a
Searcher randomizes between unit speed paths on a network, aiming to find a
hidden point in minimal expected time in the worst case. This can be viewed as
a zero-sum game between the Searcher and a time maximizing Hider. It is a
natural model for many search problems such as search and rescue operations;
the search for an enemy, a bomb or weapons in a military context; or
predator-prey search. A Chinese Postman Tour (CPT) is a minimal time tour of
the network that searches all the arcs and a Random Chinese Postman Tour (RCPT)
is an equiprobable choice of any given CPT and its reverse. The full class of
networks for which a RCPT is optimal is known, but otherwise little is known
about the solution of the game except in some special cases that have
complicated optimal strategies that would be impractical to implement. The
question of how well a RCPT or any other search strategy performs for general
networks has never been analyzed. We show that a RCPT has an approximation
ratio of $4/3$: that is, the maximum expected time it takes to find a point on
the network is no greater than $4/3$ times that of the optimal search strategy.
We then examine the performance of a RCPT in a related search game recently
proposed by S. Alpern in which the Searcher must return to his starting point
after finding the Hider.
