Sustained Vertex Cover on Temporal Graphs

We consider a novel vertex cover problem on temporal graphs, where the edges in the graph may change over time, and a vertex selected into the solution has a lifespan d. Specifically, a vertex selected at time t can cover all incident edges in graphs from time slot t to t+d-1. This model effectively captures the scenario of monitoring communication links via secure nodes (monitors) with limited lifespan in a dynamic network. We provide a systematic study of this problem from both theoretical and practical perspectives. We analyze its computational complexity, develop approximation and online algorithms with tight ratios, and present a parameterized algorithm and a tight quadratic kernel under fixed d. Experimental results on random and real-world temporal networks demonstrate the effectiveness of our algorithms. We believe that our systematic study not only reveals the nature of the problem itself, but also paves the way for investigating the ''sustained'' version of other problems on temporal graphs.