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).
CITATION STYLE
Banik, A., & Choudhary, P. (2018). Fixed-parameter tractable algorithms for tracking set problems. In Communications in Computer and Information Science (Vol. 10743 LNCS, pp. 93–104). Springer Verlag. https://doi.org/10.1007/978-3-319-74180-2_8
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