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Fixed-parameter algorithms for DAG partitioning. (English) Zbl 1355.05204
Summary: Finding the origin of short phrases propagating through the web has been formalized by J. Leskovec et al. [“Meme-tracking and the dynamics of the news cycle”, in: Proceedings of the 15th ACM SIGKDD international conference on knowledge discovery and data mining, KDD’09, Paris, France, June 28 – July 01, 2009, New York, NY: Association for Computing Machinery (ACM). 497–506 (2009; doi:10.1145/1557019.1557077)] 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 $$O(2^k \cdot(n+m))$$ 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 $$k \leq 190$$ 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 $$2^{O(t^2)} \cdot n$$ time if a width-$$t$$ tree decomposition of the input graph is given. Thus, we improve an algorithm and answer an open question of S. Alamdari and A. Mehrabian [Lect. Notes Comput. Sci. 7323, 17–28 (2012; Zbl 1342.05109)].
We complement our algorithms by lower bounds on the running time of exact algorithms and on the effectivity of data reduction.

##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C85 Graph algorithms (graph-theoretic aspects) 05C82 Small world graphs, complex networks (graph-theoretic aspects) 05C12 Distance in graphs
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