René van Bevern, Robert Bredereck, Morgan Chopin, Sepp Hartung, Falk Hüffner, André Nichterlein, and Ondřej Suchý. Fixed-parameter algorithms for DAG partitioning. Discrete Applied Mathematics, 220:134–160, 2017.

Abstract Finding the origin of short phrases propagating through the web has been formalized by Leskovec et al. (2009) as {DAG} Partitioning: given an arc-weighted directed acyclic graph on n  vertices and m  arcs, delete arcs with total weight at most  k such that each resulting weakly-connected component contains exactly one sink—a vertex without outgoing arcs. {DAG} Partitioning is NP-hard. We show an algorithm to solve {DAG} Partitioning in O ( 2 k ⋅ ( n + m ) )  time, that is, in linear time for fixed  k . We complement it with linear-time executable data reduction rules. Our experiments show that, in combination, they can optimally solve {DAG} Partitioning on simulated citation networks within five minutes for k ≤ 190 and m being  10 7 and larger. We use our obtained optimal solutions to evaluate the solution quality of Leskovec et al.’s heuristic. We show that Leskovec et al.’s heuristic works optimally on trees and generalize this result by showing that {DAG} Partitioning is solvable in 2 O ( t 2 ) ⋅ n time if a width- t tree decomposition of the input graph is given. Thus, we improve an algorithm and answer an open question of Alamdari and Mehrabian (2012). We complement our algorithms by lower bounds on the running time of exact algorithms and on the effectivity of data reduction.

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