Minimizing the average searching time for an object within a graph

This paper presents a new graph search problem for which a searcher wishes to find an object that may be found at a set of locations. The searcher doesn’t know the object’s exact location, but does know the a-prior probability of finding the object at each location. He wishes to build a searching path for reaching the object that starts from a given location and ends when reaching the object (or after searching the entire set with a false result). The objective is to find a searching path which will minimize the average searching time. We consider two scenarios for this problem: one when there is an unknown number of objects on the set and another when there is exactly one object on the set (the sum of probabilities is equal to 1). We show that this problem is NP-Hard, and supply a branch and bound algorithm for finding an optimal solution for large scale problems. We also study greedy approaches and other heuristics and compare the performance of these algorithms in various situations.