## 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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