Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2006-05

March 2006

Searching Symmetric Networks with Utilitarian Postman Paths

Steve Alpern, Vic Baston and Shmuel Gal

For any network Q, one may consider the zero-sum search game G(Q)  in which the (minimizing) Searcher picks a unit speed path S(t) in Q, the Hider picks a point  H in Q and the payoff is the meeting time T = min {t: S(t)=H}.:We show first that if Q is symmetric (edge and vertex transitive), then it is optimal for the Hider to pick H  uniformly in Q, so that the Searcher must follow a Utilitarian Postman path (one which minimizes the time to reach a random point).  We then show that if  Q is symmetric of odd degree,  with  n vertices and  m unit length edges, the value  V of G(Q) satisfies V>=m/2 + (n-2n), with equality if and only if  Q  has a path e(1),...,e(n-2) which includes  n-1 vertices such that Q - {e(2), e(4), ... , e(n-2) is a connected set of edges.

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