Fixed-parameter tractable algorithms for Tracking Shortest Paths

by   Aritra Banik, et al.

We consider the parameterized complexity of the problem of tracking shortest s-t paths in graphs, motivated by applications in security and wireless networks. Given an undirected and unweighted graph with a source s and a destination t, Tracking Shortest Paths asks if there exists a k-sized subset of vertices (referred to as tracking set) that intersects each shortest s-t path in a distinct set of vertices. We first generalize this problem for set systems, namely Tracking Set System, where given a family of subsets of a universe, we are required to find a subset of elements from the universe that has a unique intersection with each set in the family. Tracking Set System is shown to be fixed-parameter tractable due to its relation with a known problem, Test Cover. By a reduction to the well-studied d-hitting set problem, we give a polynomial (with respect to k) kernel for the case when the set sizes are bounded by d. This also helps solving Tracking Shortest Paths when the input graph diameter is bounded by d. While the results for Tracking Set System help to show that Tracking Shortest Paths is fixed-parameter tractable, we also give an independent algorithm by using some preprocessing rules, resulting in an improved running time.



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1 Introduction and Motivation

In this paper, we consider the parameterized complexity of the problem of tracking shortest - paths in graphs and some related versions of the problem. Given a graph with a specified source and a destination , a simple path between and is referred to as an - path, and a shortest simple path between and is referred to as a shortest - path. In Tracking Shortest Paths problems, the goal is to find a small subset of vertices that can help uniquely identify all shortest - paths in a graph.

We start with some motivation for the problem. Consider the security system at a large airport. As a security measure it is required to identify the route taken by passengers across the airport from entry to departure or from arrival to exit. A set of carefully chosen security scan points can be selected as identification points to trace the movement of passengers. A similar scenario can arise in any other secure facility where movement of entities need to be tracked. Note that in practical scenarios, often it is resourceful to use the shortest - paths available.

Other major application scenarios are tracking of moving objects in telecommunication networks and road networks. The goal can be efficient and optimized tracking of objects, for the purpose of surveillance, monitoring, intruder detection, and operations management. The solution to the problem can then be used for reconstruction of path traced by an object in order to detect potential network flaws, to study traffic patterns of moving objects, to optimize network resources based on such patterns, and for other such network analysis based tasks.

Tracking of moving objects has been studied in the field of wireless sensor networks. See survey for a survey of target tracking protocols using wireless sensor networks. Some researchers have studied this with respect to power management of sensors power

. Despite being an active area of research, a major part of this research so far has been based on heuristics. In 

banik, the authors formalized the problem of tracking in networks as a graph theoretic problem and did a systematic study. Among other problems, they introduced the following optimization problem. is used to denote the set of vertices in path . A graph with a unique source and unique destination is called an - graph.

Problem 1

Tracking Shortest PathsAn undirected - graph .A minimum set of vertices , such that for any two distinct shortest - paths and in , it holds that .

The output set of vertices is referred to as a tracking set and the vertices in a tracking set are called trackers. In banik, Tracking Shortest Paths was shown to be NP-hard for undirected graphs and a -approximate algorithm was given for the case of planar graphs. An -approximation algorithm for a minimization problem gives a solution that is at most times the size of an optimum solution, in time polynomial in the input size.

Tracking Shortest Paths can be generalized to the case where not just the shortest - paths, but all - paths in a graph need to be identified uniquely by a minimum subset of vertices. For a set of vertices and a path , denotes the sequence in which the vertices from appear in path . Formally the problem of tracking all - paths in a graph is defined as follows