Fixed-Parameter Algorithms for DAG Partitioning

TL;DR

This paper introduces fixed-parameter algorithms for DAG Partitioning, enabling optimal solutions for large graphs with small parameter k, and evaluates heuristic quality using these solutions.

Contribution

It presents an $O(2^k imes (n+m))$ time algorithm for DAG Partitioning, along with data reduction rules, and extends solutions to graphs with bounded treewidth, answering open questions.

Findings

Algorithm solves DAG Partitioning efficiently for small k.

Data reduction rules improve practical solving times.

Heuristic performs optimally on trees and is evaluated against optimal solutions.

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 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 $k\leq190$ 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.