In this work, we study the problem of clearing contamination spreading through a large network where we model the problem as a graph searching game. The problem can be summarized as constructing a search strategy that will leave the graph clear of any contamination at the end of the searching process in as few steps as possible. We show that this problem is NP-hard even on directed acyclic graphs and provide an efficient approximation algorithm. We experimentally observe the performance of our approximation algorithm in relation to the lower bound on several large online networks including Slashdot, Epinions and Twitter. The experiments reveal that in most cases our algorithm performs near optimally.
Graph searching is the game of capturing a fugitive by a team of searchers in a network. There are equivalent characterizations in terms of path-width, interval thickness, and vertex separation. So far the interest has mainly focused on the search number of a graph, which is the minimal the number of searchers to win the game, and accordingly on the width and the thickness. These parameters measure the needed resources and correspond to space complexity. As its dual, we introduce the search time, which has not yet been studied in graph searching. We prove that all main results on graph searching can be generalized to include search time, such as monotone or recontamination free graph searching, and the characterizations in terms of path-width, interval graphs, and vertex separation, for which we introduce appropriate length parameters. We establish the NP-completeness of both search-width and search-time. Finally we investigate the speed-up by an extra searcher. There are ’good’ classes of graphs where a single extra searcher reduces the search time to one half and ’bad’ ones where some extra searchers are no real help.