Fixed-parameter tractable algorithms for tracking set problems

We consider parameterized complexity of the recently introduced problem of tracking paths in graphs, motivated by applications in security and wireless networks. Given an undirected and unweighted graph with a specified source s and a terminal t, the goal is to find a k-sized subset of vertices that intersect with each s-t path (or s-t shortest) path in a distinct sequence (or set). We first generalize this problem to a problem on set systems with a universe of size n and a m sized family of subsets of the universe. Using a correspondence with the well-studied Test Cover Problem, we give a lower bound of lg m for the solution size and show the problem fixed-parameter tractable. We also show that when k is the parameter, then for such a set system finding a Tracking Set for such a set system of size at most lg m+k is hard for parameterized complexity class W[2];finding a Tracking Set of size at most m-k is fixed parameter tractable;finding a Tracking Set of size at most n-k is complete for parameterized complexity class W[1]. Using the solution for the set system generalization, we show the main result of the paper that finding a Tracking Set of size at most k for shortest paths is fixed-parameter tractable. We first give an (Formula presented) algorithm using the set system solution, which we later improve to (Formula presented).