Average Sensitivity of Graph Algorithms
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In modern applications of graphs algorithms, where the graphs of interest are large and dynamic, it is unrealistic to assume that an input representation contains the full information of a graph being studied. Hence, it is desirable to use algorithms that, even when only a (large) subgraph is available, output solutions that are close to the solutions output when the whole graph is available. We formalize this idea by introducing the notion of average sensitivity of graph algorithms, which is the average earth mover's distance between the output distributions of an algorithm on a graph and its subgraph obtained by removing an edge, where the average is over the edges removed and the distance between two outputs is the Hamming distance. In this work, we initiate a systematic study of average sensitivity. After deriving basic properties of average sensitivity such as composability, we provide efficient approximation algorithms with low average sensitivities for concrete graph problems, including the minimum spanning forest problem, the global minimum cut problem, the maximum matching problem, and the minimum vertex cover problem. We also show that every algorithm for the 2-coloring problem has average sensitivity linear in the number of vertices. To show our algorithmic results, we establish and utilize the following fact; if the presence of a vertex or an edge in the solution output by an algorithm can be decided locally, then the algorithm has a low average sensitivity, allowing us to reuse the analyses of known sublinear-time algorithms.
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