Querying with Uncertainty

Abstract

We introduce and study a new optimization problem on querying with uncertainty. k robots are required to locate a hidden item that is placed uniformly at random in one of n different locations, each associated with a probability $p_i$, $i=1,\ldots ,n$. If the item is placed in location i, a query trial by any of the robots reveals the item with probability $p_i$. Each robot j is assigned a subset $A_j$ of the locations, and is allowed to perform a random walk among them, each time step querying the current location (being visited) for the item. We are interested in determining sets $\{A_j\}_{j=1,\ldots ,k}$ so as to minimize the expected discovery time of the item. We measure the cost by the number of queries, while there is no cost for hopping from node to node.

Our first contribution is to prove a closed formula for the expected number of steps until the treasure is found when the robots execute unanimous queries. Then we focus on querying problems where the sets $A_j$ are restricted to be either pairwise disjoint or identical. Our findings allow us to obtain optimal solutions, when sets $A_j$ are exclusively pairwise disjoint, requiring time $n^{O(k)}$. In our second contribution, we devise an optimal polynomial time algorithm for querying with $k=2$ robots even when the sets $A_1,A_2$ are allowed to overlap. All our algorithms are based on special concavity-type properties of the expected termination time when the robots execute unanimous queries, thus inducing special structural properties of optimal solutions for the general problem.

K. Georgiou and E. Kranakis—Research supported in part by NSERC Discovery grant.