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.
The notion of search time for undirected graphs was introduced by Brandenburg and Herrmann @cite_13 . They note that the classical goal of the graph searching game where the minimal search number is computed aims to minimize the number of resources used and as such corresponds to space complexity. They study the length of a search strategy which corresponds to the time complexity of searching a graph. They ask, how fast can a team of @math searchers clear a graph (if at all), and conversely how many searchers are needed to search a graph in time @math .
@cite_13
1566624605 2053218689 105334049 2094887806
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. In the graph searching game the opponents are a set of searchers and a fugitive in a graph. The searchers try to capture the fugitive by applying some sequence of moves that include placement, removal, or sliding of a searcher along an edge. The fugitive tries to avoid capture by moving along unguarded paths. The search number of a graph is the minimum number of searchers required to guarantee the capture of the fugitive. In this paper, we initiate the study of this game under the natural restriction of connectivity where we demand that in each step of the search the locations of the graph that are clean (i.e. non-accessible to the fugitive) remain connected. We give evidence that many of the standard mathematical tools used so far in classic graph searching fail under the connectivity requirement. We also settle the question on ''the price of connectivity'', that is, how many searchers more are required for searching a graph when the connectivity demand is imposed. We make estimations of the price of connectivity on general graphs and we provide tight bounds for the case of trees. In particular, for an n-vertex graph the ratio between the connected searching number and the non-connected one is O(logn) while for trees this ratio is always at most 2. We also conjecture that this constant-ratio upper bound for trees holds also for all graphs. Our combinatorial results imply a complete characterization of connected graph searching on trees. It is based on a forbidden-graph characterization of the connected search number. We prove that the connected search game is monotone for trees, i.e. restricting search strategies to only those where the clean territories increase monotonically does not require more searchers. A consequence of our results is that the connected search number can be computed in polynomial time on trees, moreover, we show how to make this algorithm distributed. Finally, we reveal connections of this parameter to other invariants on trees such as the Horton-Strahler number. We study the problem of exploring an unknown undirected connected graph. Beginning in some start vertex, a searcher must visit each node of the graph by traversing edges. Upon visiting a vertex for the first time, the searcher learns all incident edges and their respective traversal costs. The goal is to find a tour of minimum total cost. Kalyanasundaram and Pruhs (Constructing competitive tours from local information, Theoretical Computer Science 130, pp. 125-138, 1994) proposed a sophisticated generalization of a Depth First Search that is 16-competitive on planar graphs. While the algorithm is feasible on arbitrary graphs, the question whether it has constant competitive ratio in general has remained open. Our main result is an involved lower bound construction that answers this question negatively. On the positive side, we prove that the algorithm has constant competitive ratio on any class of graphs with bounded genus. Furthermore, we provide a constant competitive algorithm for general graphs with a bounded number of distinct weights. [20th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1979, pp. 218--223] posed the following question: "The reachability problem for undirected graphs can be solved in log space and @math time [ @math is the number of edges and @math is the number of vertices] by a probabilistic algorithm that simulates a random walk, or in linear time and space by a conventional deterministic graph traversal algorithm. Is there a spectrum of time-space trade-offs between these extremes?" This question is answered in the affirmative for sparse graphs by presentation of an algorithm that is faster than the random walk by a factor essentially proportional to the size of its workspace. For denser graphs, this algorithm is faster than the random walk but the speed-up factor is smaller.