The covert set-cover problem with application to Network Discovery Sandeep Sen and V.N. Muralidhara Department of Computer Science and Engineering, Indian Institute of Technology, Delhi, India. {ssen,murali}@cse.iitd.ernet.in Abstract. We address a version of the set-cover problem where we do not know the sets initially (and hence referred to as covert) but we can query an element to find out which sets contain this element as well as query a set to know the elements. We want to find a small set-cover using a minimal number of such queries. We present a Monte Carlo randomized algorithm that approximates an optimal set-cover of size OPT within O(log N ) factor with high probability using O(OPT · log 2 N ) queries where N is the number of element in the universal set. We apply this technique to the network discovery problem that involves certifying all the edges and non-edges of an unknown n-vertices graph based on layered-graph queries from a minimal number of vertices. By reducing it to the covert set-cover problem we present an O(log 2 n)- competitive Monte Carlo randomized algorithm for the covert version of network discovery problem. The previously best known algorithm has a competitive ratio of Ω( √ n log n) and therefore our result achieves an exponential improvement. 1 Introduction Given a ground set S with n ′ elements and a family of sets S 1 ,S 2 ...S m ′ where 1 S i ⊂ S,a cover C is a collection of sets from this family whose union is S. It is known that finding a cover consisting of the minimum number of sets is a computationally intractable problem [9]. There are many strategies [11, 6, 10] to approximate the smallest cover within a factor of O(log n ′ ) which is known to be the best possible unless P = NP [7]. In this paper, we consider the following version of the set cover problem. Although we know m ′ ,n ′ , we do not know the elements nor the cardinality of any of the sets S i . We are allowed to query an element e ∈ S that returns all sets S i that contain e which we refer to as a hitting-set query; we can also query a set to know its elements. We would like to compute a small set cover of S using a minimal number of such queries. More specifically, if OPT is the minimum size of a set cover for any instance of the problem, we would like to find a set 1 We have chosen n ′ ,m ′ as notations to keep them distinct from graphs with n vertices and m edges.