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On the Complexity of the Bilevel Shortest Path Problem

Dorothee Henke, Lasse Wulf

TL;DR

The paper studies BSP, a bilevel variant of the shortest path where the edge set is partitioned between a leader and a follower, and analyzes both strong and weak path completion across directed, undirected, and directed acyclic graphs. It delivers a complete complexity classification, showing $P$, NP-complete, or $\Sigma^p_2$-complete results for each variant, and introduces a vertex fixing lemma and a Min-Max-Ham reduction that establish $\Sigma^p_2$-completeness for the leader in the strong variant. For the weak variant, the follower’s problem reduces to a standard shortest path and is polynomial-time solvable, while the strong variant remains hard; in DAGs, the strong variant is tractable via dynamic programming. In the few-leader-edge regime the problem becomes equivalent to Shortest-${k}$-Cycle, linking BSP to a known combinatorial problem with nuanced complexity status and implying randomized algorithms with potential derandomization contingent on Shortest-${k}$-Cycle progress.

Abstract

We introduce a new bilevel version of the classic shortest path problem and completely characterize its computational complexity with respect to several problem variants. In our problem, the leader and the follower each control a subset of the edges of a graph and together aim at building a path between two given vertices, while each of the two players minimizes the cost of the resulting path according to their own cost function. We investigate both directed and undirected graphs, as well as the special case of directed acyclic graphs. Moreover, we distinguish two versions of the follower's problem: Either they have to complete the edge set selected by the leader such that the joint solution is exactly a path, or they have to complete the edge set selected by the leader such that the joint solution is a superset of a path. In general, the bilevel problem turns out to be much harder in the former case: We show that the follower's problem is already NP-hard here and that the leader's problem is even hard for the second level of the polynomial hierarchy, while both problems are one level easier in the latter case. Interestingly, for directed acyclic graphs, this difference turns around, as we give a polynomial-time algorithm for the first version of the bilevel problem, but it stays NP-hard in the second case. Finally, we consider restrictions that render the problem tractable. We prove that, for a constant number of leader's edges, one of our problem variants is actually equivalent to the shortest-$k$-cycle problem, which is a known combinatorial problem with partially unresolved complexity status. In particular, our problem admits a polynomial-time randomized algorithm that can be derandomized if and only if the shortest-$k$-cycle problem admits a deterministic polynomial-time algorithm.

On the Complexity of the Bilevel Shortest Path Problem

TL;DR

The paper studies BSP, a bilevel variant of the shortest path where the edge set is partitioned between a leader and a follower, and analyzes both strong and weak path completion across directed, undirected, and directed acyclic graphs. It delivers a complete complexity classification, showing , NP-complete, or -complete results for each variant, and introduces a vertex fixing lemma and a Min-Max-Ham reduction that establish -completeness for the leader in the strong variant. For the weak variant, the follower’s problem reduces to a standard shortest path and is polynomial-time solvable, while the strong variant remains hard; in DAGs, the strong variant is tractable via dynamic programming. In the few-leader-edge regime the problem becomes equivalent to Shortest--Cycle, linking BSP to a known combinatorial problem with nuanced complexity status and implying randomized algorithms with potential derandomization contingent on Shortest--Cycle progress.

Abstract

We introduce a new bilevel version of the classic shortest path problem and completely characterize its computational complexity with respect to several problem variants. In our problem, the leader and the follower each control a subset of the edges of a graph and together aim at building a path between two given vertices, while each of the two players minimizes the cost of the resulting path according to their own cost function. We investigate both directed and undirected graphs, as well as the special case of directed acyclic graphs. Moreover, we distinguish two versions of the follower's problem: Either they have to complete the edge set selected by the leader such that the joint solution is exactly a path, or they have to complete the edge set selected by the leader such that the joint solution is a superset of a path. In general, the bilevel problem turns out to be much harder in the former case: We show that the follower's problem is already NP-hard here and that the leader's problem is even hard for the second level of the polynomial hierarchy, while both problems are one level easier in the latter case. Interestingly, for directed acyclic graphs, this difference turns around, as we give a polynomial-time algorithm for the first version of the bilevel problem, but it stays NP-hard in the second case. Finally, we consider restrictions that render the problem tractable. We prove that, for a constant number of leader's edges, one of our problem variants is actually equivalent to the shortest--cycle problem, which is a known combinatorial problem with partially unresolved complexity status. In particular, our problem admits a polynomial-time randomized algorithm that can be derandomized if and only if the shortest--cycle problem admits a deterministic polynomial-time algorithm.
Paper Structure (14 sections, 15 theorems, 12 equations, 5 figures, 2 tables)

This paper contains 14 sections, 15 theorems, 12 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related literature
  3. Problem formulation
  4. General insights
  5. Weak path completion variant
  6. Strong path completion variant
  7. Vertex fixing lemma
  8. NP-completeness of the follower's problem
  9. $\Sigma^p_2$-completeness of the leader's problem
  10. Polynomial-time algorithms for directed acyclic graphs
  11. Case of few leader's edges
  12. Equivalence to the shortest-$k$-cycle problem
  13. Inapproximability
  14. Conclusion

Key Result

Lemma 1

For both the strong and the weak path completion variant of the Bilevel Shortest Path problem, the undirected case can be polynomially reduced to the directed case.

Figures (5)

  • Figure 1: Illustration of the construction in the proof of \ref{['lem:undir-special-case-of-dir']} for the reduction of the undirected case to the directed case.
  • Figure 2: Instance discussed in \ref{['ex:weak-vs-strong']}. The leader's costs and edges are highlighted in blue, the follower's costs and edges are highlighted in red. Colored figure available online.
  • Figure 3: Construction used in the proof of \ref{['thm:BSP-weak-NP-complete']}. Leader's edges and costs are marked in blue, and follower's edges and costs are marked in red. Colored figure available online.
  • Figure 4: Construction used in \ref{['lem:lifting-lemma']}. The table explains the meaning of the different edge types. Colored figure available online.
  • Figure 5: Example of the path contraction used in the proof of \ref{['lem:lifting-lemma']}.

Theorems & Definitions (31)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Example 3
  • Lemma 4
  • proof
  • Corollary 6
  • proof
  • Theorem 7
  • ...and 21 more