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

Nour ElHouda Tellache, Roberto Baldacci

TL;DR

This paper studies MPC-ARC, a variant of the Minimum Path Cover on DAGs where every path must include an arc from a given subset \widecheck{A}. It proves strong NP-hardness for the feasibility problem on general DAGs and identifies a polynomial-time case for transitive closures of paths, linking to airline crew scheduling constraints. It then develops two integer programming formulations for the maximum-coverage variant MFC-ARC, enhances them with a broad suite of valid inequalities (including reachability and generalized cut constraints), and embeds these in a branch-and-cut framework with specialized separation routines. Extensive computational experiments, including real airline scheduling instances, demonstrate the effectiveness of the approach and highlight the impact of DAG structure and \widecheck{A} sparsity on performance. The work advances exact methods for constrained path-cover problems with practical significance in scheduling and related applications.

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.

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

TL;DR

This paper studies MPC-ARC, a variant of the Minimum Path Cover on DAGs where every path must include an arc from a given subset \widecheck{A}. It proves strong NP-hardness for the feasibility problem on general DAGs and identifies a polynomial-time case for transitive closures of paths, linking to airline crew scheduling constraints. It then develops two integer programming formulations for the maximum-coverage variant MFC-ARC, enhances them with a broad suite of valid inequalities (including reachability and generalized cut constraints), and embeds these in a branch-and-cut framework with specialized separation routines. Extensive computational experiments, including real airline scheduling instances, demonstrate the effectiveness of the approach and highlight the impact of DAG structure and \widecheck{A} sparsity on performance. The work advances exact methods for constrained path-cover problems with practical significance in scheduling and related applications.

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.
Paper Structure (19 sections, 13 theorems, 15 equations, 4 figures, 51 tables)

This paper contains 19 sections, 13 theorems, 15 equations, 4 figures, 51 tables.

Table of Contents

  1. Introduction
  2. Complexity results
  3. Path-flow based mathematical formulations
  4. A three-index path flow formulation
  5. A two-index path flow formulation
  6. Classes of valid inequalities
  7. Tournament constraints
  8. $\widecheck{A}$-reachability constraints
  9. Reachability cuts
  10. Generalized cut constraints
  11. Exact branch-and-cut algorithm
  12. Separation routines
  13. Separation, branching, and node selection strategies
  14. Computational experiments
  15. Instances
  16. ...and 4 more sections

Key Result

Theorem 1

FPC-ARC is $\mathcal{NP}$-hard in the strong sense for arbitrary DAGs.

Figures (4)

  • Figure 1: Gadget $G_{m}=(V_{m},A_{m})$
  • Figure 2: Path of two different gadgets $G_{m_1}$ and $G_{m_2}$
  • Figure 3: Feasible cases
  • Figure 4: Infeasible cases

Theorems & Definitions (22)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Proposition 1: Trivial Constraint (TIC)
  • Proposition 2: Tournament Constraints I (TC-I)
  • proof
  • Proposition 3: Tournament Constraints II (TC-II)
  • proof
  • Proposition 4
  • ...and 12 more