Minimum Path Cover: The Power of Parameterization

TL;DR

This paper presents the first high-performance implementation of advanced algorithms for computing minimum path covers in DAGs, demonstrating significant speedups especially on dense graphs and introducing effective heuristics.

Contribution

It provides the first publicly available high-performance implementation of MPC algorithms, including parameterized approaches, and introduces new heuristics for improved performance.

Findings

Parameterized algorithms are orders-of-magnitude faster on dense graphs.

Heuristics based on transitive edge sparsification significantly improve solver performance.

Experimental results confirm the efficiency of the proposed methods.

Abstract

Computing a minimum path cover (MPC) of a directed acyclic graph (DAG) is a fundamental problem with a myriad of applications, including reachability. Although it is known how to solve the problem by a simple reduction to minimum flow, recent theoretical advances exploit this idea to obtain algorithms parameterized by the number of paths of an MPC, known as the width. These results obtain fast [M\"akinen et al., TALG] and even linear time [C\'aceres et al., SODA 2022] algorithms in the small-width regime. In this paper, we present the first publicly available high-performance implementation of state-of-the-art MPC algorithms, including the parameterized approaches. Our experiments on random DAGs show that parameterized algorithms are orders-of-magnitude faster on dense graphs. Additionally, we present new pre-processing heuristics based on transitive edge sparsification. We show that our heuristics improve MPC-solvers by orders-of-magnitude.