A Combinatorial Model of Two-Sided Search

We study a new model of combinatorial group testing: the item to be found (a.k.a. the target) occupies an unknown node in a graph. At each time instant, we can test (or query) a subset of the nodes and learn whether the target occupies any of such nodes. Immediately after the result of the test is available, the target can move to any node adjacent to its present location. The search finishes when we are able to locate the object with some predefined accuracy s (a parameter fixed beforehand), i.e., to indicate a set of s nodes that includes the location of the object. In this paper we study two types of problems related to the above model: (i) what is the minimum value of the accuracy parameter for which a search strategy in the above sense exists; (ii) given the accuracy, what is the minimum number of tests that allow to locate the target. We study these questions on paths, cycles, and trees as underlying graphs and provide tight answers for the above questions. We also consider a restricted variant of the problem, where the number of moves of the target is bounded.