The thinnest path problem for secure communications: A directed hypergraph approach

We formulate and study the thinnest path problem in wireless ad hoc networks. The objective is to find a path from the source to the destination that results in the minimum number of nodes overhearing the message by carefully choosing the relaying nodes and their corresponding transmission power. We adopt a directed hypergraph model of the problem and establish the NP-completeness of the problem in 2-D networks. We then develop a polynomial-time approximation algorithm that offers a √n/2 approximation ratio for general directed hypergraphs (which can model non-isomorphic signal propagation in space) and constant approximation ratio for disk hypergraphs (which result from isomorphic signal propagation). We also consider the thinnest path problem in 1-D networks and 1-D networks embedded in 2-D field of eavesdroppers with arbitrary unknown locations (the so-called 1.5-D networks). We propose a linear-complexity algorithm based on nested backward induction that obtains the optimal solution to both 1-D and 1.5-D networks. In particular, no algorithm, even with the complete knowledge of the locations of the eavesdroppers, can obtain a thinner path than the proposed algorithm which does not require the knowledge of eavesdropper locations.