FixedParameter Algorithms for DAG Partitioning
Abstract
Finding the origin of short phrases propagating through the web has been formalized by Leskovec et al. [ACM SIGKDD 2009] as DAG Partitioning: given an arcweighted directed acyclic graph on $n$ vertices and $m$ arcs, delete arcs with total weight at most $k$ such that each resulting weaklyconnected component contains exactly one sinka vertex without outgoing arcs. DAG Partitioning is NPhard. 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 lineartime executable data reduction rules. Our experiments show that, in combination, they can optimally solve DAG Partitioning on simulated citation networks within five minutes for $k\leq190$ 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 $2^{O(w^2)}\cdot n$ time if a width$w$ tree decomposition of the input graph is given. Thus, we improve an algorithm and answer an open question of Alamdari and Mehrabian [WAW 2012]. We complement our algorithms by lower bounds on the running time of exact algorithms and on the effectivity of data reduction.
 Publication:

arXiv eprints
 Pub Date:
 November 2016
 arXiv:
 arXiv:1611.08809
 Bibcode:
 2016arXiv161108809V
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Discrete Mathematics;
 Computer Science  Social and Information Networks;
 F.2.2;
 G.2.2
 EPrint:
 A preliminary version of this article appeared at CIAC'13. Besides providing full proof details, this revised and extended version improves our O(2^k * n^2)time algorithm to run in O(2^k * (n+m)) time and provides lineartime executable data reduction rules. Moreover, we experimentally evaluated the algorithm and compared it to known heuristics