On the Hardness of Eliminating Cheating Behavior in Time Synchronization Protocols for Sensor Networks

Wireless Sensor Networks are fast gaining popularity for use in a variety of remote sensing, emergency monitoring and information collection applications. Time synchronization in such highly distributed systems is important in order to maintain a global notion of time throughout the network and to support the underlying applications. But, cheating behavior by malicious nodes can severely jeopardize the accuracy of the time synchronization service associated with the network and the related time-based applications. Existing literature on the problem of detection and elimination of cheating behavior in distributed time synchronization protocols has very little or no fundamental theoretical analysis of the basic problem itself. Absence of sound theoretical models and analysis of time synchronization protocols in such networks has left a number of questions unanswered: How hard is it to detect and eliminate cheating or inconsistent behavior in time synchronization protocols given complete (local time and time difference) information? Do efficient algorithms for doing the same exist and if they do, is there a performance guarantee on the solution quality? In this paper, we attempt to answer these questions. We first present a practical graph-theoretic model, called Time Difference Graphs (TDG), for the network and formulate the time synchronization problem as a Constraint Satisfaction Problem (CSP) in such graphs. We then show that efficiently eliminating cheating behavior (or inconsistency causing nodes) in TDGs is NP-hard. Furthermore, we show that this problem is hard even for a special case of TDG, namely, a completely connected TDG. Moreover, we show that it is impossible to approximate this problem within a factor n1−ε, where ε > 0, unless P = NP. Finally, we propose two polynomial time approximation strategies, namely a greedy strategy and an Integer Programming formulation, for efficiently eliminating inconsistency causing nodes in complete TDGs.