TY - JOUR

T1 - Fixed-parameter algorithms for DAG Partitioning

AU - van Bevern, René

AU - Bredereck, Robert

AU - Chopin, Morgan

AU - Hartung, Sepp

AU - Hüffner, Falk

AU - Nichterlein, André

AU - Suchý, Ondřej

N1 - Publisher Copyright:
© 2016 Elsevier B.V.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2017/3/31

Y1 - 2017/3/31

N2 - 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(2k⋅(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 107 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 2O(t2)⋅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.

AB - 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(2k⋅(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 107 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 2O(t2)⋅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.

KW - Algorithm engineering

KW - Evaluating heuristics

KW - Graph algorithms

KW - Linear-time algorithms

KW - Multiway cut

KW - NP-hard problem

KW - Polynomial-time data reduction

KW - KERNELIZATION

KW - MULTIVARIATE ALGORITHMICS

KW - COMPLEXITY

UR - http://www.scopus.com/inward/record.url?scp=85008685640&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2016.12.002

DO - 10.1016/j.dam.2016.12.002

M3 - Article

AN - SCOPUS:85008685640

VL - 220

SP - 134

EP - 160

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

ER -