Counting and localizing defective nodes by Boolean network tomography

Identifying defective items in larger sets is a main problem with many applications in real life situations. We consider the problem of localizing defective nodes in networks through an approach based on boolean network tomography (BNT), which is grounded on inferring informations from the boolean outcomes of end-to-end measurements paths. {\em Identifiability} conditions on the set of paths which guarantee discovering or counting unambiguously the defective nodes are of course very relevant. We investigate old and introduce new identifiability conditions contributing this problem both from a theoretical and applied perspective. (1) What is the precise tradeoff between number of nodes and number of paths such that at most $k$ nodes can be identified unambiguously ? The answer is known only for $k=1$ and we answer the question for any $k$, setting a problem implicitly left open in previous works. (2) We study upper and lower bounds on the number of unambiguously identifiable nodes, introducing new identifiability conditions which strictly imply and are strictly implied by unambiguous identifiability; (3) We use these new conditions on one side to design algorithmic heuristics to count defective nodes in a fine-grained way, on the other side to prove the first complexity hardness results on the problem of identifying defective nodes in networks via BNT. (4) We introduce a random model where we study lower bounds on the number of unambiguously identifiable defective nodes and we use this model to estimate that number on real networks by a maximum likelihood estimate approach