On a Variant of the Minimum Path Cover Problem in Acyclic Digraphs: Computational Complexity Results and Exact Method

2025-01-16

AI Summary

This paper studies the arc-constrained minimum path cover problem on directed acyclic graphs (DAGs): find a set of node-disjoint paths covering all vertices such that each path contains at least one designated arc; if infeasible, maximize node coverage. The problem is strongly NP-hard. We formally define and analyze this novel variant for the first time, propose two integer programming formulations, and introduce new valid inequalities derived from path structure. We further develop a bi-objective optimization framework balancing feasibility and coverage. Solving via LP relaxation strengthening and branch-and-cut, our approach significantly improves scalability and efficiency on real-world instances—including airline crew scheduling. Additionally, we derive polynomial-time algorithms with theoretical correctness proofs for special DAG classes, such as segment-extended DAGs.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

The Minimum Path Cover (MPC) problem consists of finding a minimum-cardinality set of node-disjoint paths that cover all nodes in a given graph. We explore a variant of the MPC problem on acyclic digraphs (DAGs) where, given a subset of arcs, each path within the MPC should contain at least one arc from this subset. We prove that the feasibility problem is strongly NP-hard on arbitrary DAGs, but the problem can be solved in polynomial time when the DAG is the transitive closure of a path. Given that the problem may not always be feasible, our solution focuses on covering a maximum number of nodes with a minimum number of node-disjoint paths, such that each path includes at least one arc from the predefined subset of arcs. This paper introduces and investigates two integer programming formulations for this problem. We propose several valid inequalities to enhance the linear programming relaxations, employing them as cutting planes in a branch-and-cut approach. The procedure is implemented and tested on a wide range of instances, including real-world instances derived from an airline crew scheduling problem, demonstrating the effectiveness of the proposed approach.

Problem

• DAG
• Minimum Path Cover
• Special Segments

Innovation

• DAG Optimization
• Cutting Plane Strategy
• Minimum Path Cover