A Combinatorial Model of Two-Sided Search

We study a new model of combinatorial group testing in a network. An object (the target) occupies an unknown node in the network. At each time instant, we can test (or query) a subset of the nodes to learn whether the target occupies any of such nodes. Unlike the case of conventional group testing problems on graphs, the target in our model can move immediately after each test to any node adjacent to each 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 include 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 answer for the above questions. We also considered a restricted variant of the problem, where the number of moves of the target is bounded.