an:06680977
Zbl 1355.05204
van Bevern, Ren??; Bredereck, Robert; Chopin, Morgan; Hartung, Sepp; H??ffner, Falk; Nichterlein, Andr??; Such??, Ond??ej
Fixed-parameter algorithms for DAG partitioning
EN
Discrete Appl. Math. 220, 134-160 (2017).
00362800
2017
j
05C70 05C85 05C82 05C12
NP-hard problem; graph algorithms; polynomial-time data reduction; multiway cut; linear-time algorithms; algorithm engineering; evaluating heuristics
Summary: Finding the origin of short phrases propagating through the web has been formalized by \textit{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; \url{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 \textit{S. Alamdari} and \textit{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.
Zbl 1342.05109