The goal of tracking the origin of short, distinctive phrases (memes) that propagate through the web in reaction to current events has been formalized as DAG Partitioning: given a directed acyclic graph, delete edges of minimum weight such that each resulting connected component of the underlying undirected graph contains only one sink. Motivated by NP-hardness and hardness of approximation results, we consider the parameterized complexity of this problem. We show that it can be solved in O(2^k·n^2) time, where k is the number of edge deletions, proving fixed-parameter tractability for parameter k. We then show that unless the Exponential Time Hypothesis (ETH) fails, this cannot be improved to 2^o(k)·n^O(1); further, DAG Partitioning does not have a polynomial kernel unless NP ⊆ coNP/poly. Finally, given a tree decomposition of width w, we show how to solve DAG Partitioning in 2^O(w^2)⋅n time, improving a known algorithm for the parameter pathwidth.
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