Secluded Path via Shortest Path

Abstract

We provide several new algorithmic results for the secluded path problem, specifically approximation and optimality results for the static algorithm of [3,5], and an extension (h-Memory) of it based on de Bruijn graphs, when applied to bounded degree graphs and some other special graph classes which can model wireless communication and line-of-sight settings. Our primary result is that h-Memory is a PTAS for degree-Δ unweighted, undirected graphs, providing a $\lceil{\sqrt{{\Delta+1}\over{h+1}}}\rceil$-approximation in time O(n logn); in particular, 0-Memory (i.e., static) provides a $\sqrt{\Delta+1}$ -approximation (i.e., $\epsilon=\sqrt{\Delta+1}-1$), tightening the previous analysis of this algorithm, and Δ-Memory is optimal (i.e., ε = 0), and is faster than the known optimal algorithm for this setting [3].

We also show that 0-Memory and 1-Memory give constant approximations for unit-disk graphs and planar graphs, and that an extension of h-Memory solves many other tessellation graphs. Finally, we prove that the problem is NP-hard on node-weighted graphs of degree 3.

Research was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-09-2-0053. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation here on.