Sparsifying, Shrinking and Splicing for Minimum Path Cover in Parameterized Linear Time

TL;DR

This paper introduces new parameterized algorithms for computing minimum path covers in DAGs, achieving linear time complexity and efficient parallelization, with applications to edge sparsification.

Contribution

The authors develop the first parameterized linear-time algorithms for MPC in DAGs and introduce techniques for edge sparsification that preserve the DAG's width.

Findings

Two new MPC algorithms with $O(k^2|V| ext{log}|V| + |E|)$ and $O(k^3|V| + |E|)$ running times.

A parallel MPC algorithm with $O(k^2|V| + |E|)$ steps using $O( ext{log}|V|)$ processors.

An asymptotically tight algorithm for transforming MPCs with fewer than $2|V|$ edges.

Abstract

A minimum path cover (MPC) of a directed acyclic graph (DAG) $G = (V,E)$ is a minimum-size set of paths that together cover all the vertices of the DAG. Computing an MPC is a basic polynomial problem, dating back to Dilworth's and Fulkerson's results in the 1950s. Since the size $k$ of an MPC (also known as the width) can be small in practical applications, research has also studied algorithms whose complexity is parameterized on $k$. We obtain two new MPC parameterized algorithms for DAGs running in time $O(k^2|V|\log{|V|} + |E|)$ and $O(k^3|V| + |E|)$. We also obtain a parallel algorithm running in $O(k^2|V| + |E|)$ parallel steps and using $O(\log{|V|})$ processors (in the PRAM model). Our latter two algorithms are the first solving the problem in parameterized linear time. Finally, we present an algorithm running in time $O(k^2|V|)$ for transforming any MPC to another MPC using less than $2|V|$ distinct edges, which we prove to be asymptotically tight. As such, we also obtain edge sparsification algorithms preserving the width of the DAG with the same running time as our MPC algorithms. At the core of all our algorithms we interleave the usage of three techniques: transitive sparsification, shrinking of a path cover, and the splicing of a set of paths along a given path.