On the Complexity of the Bilevel Shortest Path Problem

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.

💡 Research Summary

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The paper introduces a novel bilevel version of the classic shortest‑path problem, called the Bilevel Shortest Path (BSP) problem, and provides a complete complexity classification for its various incarnations. In the BSP setting a graph G =(V,E) is partitioned into a set Eℓ controlled by a leader and a set Ef controlled by a follower. Both players share the same source‑target pair (s,t) and together must construct an s‑t path. The leader moves first, selecting a subset X⊆Eℓ. The follower then selects a subset Y′⊆Ef to complete the solution. Each player has its own non‑negative cost function: c for the leader (applied to the whole final path X∪Y′) and d for the follower (applied only to the follower’s edges). The objective is hierarchical: the follower minimizes d(Y′) subject to feasibility, and the leader minimizes c(X∪Y′) given the follower’s optimal reaction.

Two feasibility notions are distinguished:

1. Strong Path Completion – every edge chosen by the leader must belong to the final s‑t path. The follower is forced to use all of X and can only add edges to make a valid path.
2. Weak Path Completion – the final solution only needs to contain an s‑t path as a subset; the follower may ignore some of the leader’s edges if doing so reduces its own cost.

For each of these notions the authors consider three graph families: directed graphs, undirected graphs, and directed acyclic graphs (DAGs). This yields six problem variants. For each variant they study both the follower’s decision problem (given X, find the cheapest feasible Y′) and the leader’s decision problem (choose X to minimize the leader’s total cost).

Main Complexity Results (summarized in Table 1)

• Strong Completion, General Directed/Undirected Graphs
  Follower: NP‑hard (Theorem 7). The follower must find a path that contains a prescribed set of edges, a task reminiscent of Hamiltonian‑Path.
  Leader: Σ₂^P‑complete (Theorem 10). The leader must anticipate the follower’s optimal reaction; this places the problem at the second level of the polynomial hierarchy, a rarity among combinatorial bilevel problems.

• Weak Completion, General Directed/Undirected Graphs
  Follower: solvable in polynomial time (standard shortest‑path algorithms).
  Leader: NP‑complete (Theorem 9). The leader’s problem reduces to a classic NP‑hard bilevel optimization, but does not climb to Σ₂^P.

• Directed Acyclic Graphs (DAGs)
  Strong Completion: polynomial‑time solvable (Theorem 14). The absence of cycles allows a topological‑order‑based dynamic program to enforce the leader’s edges. Consequently the leader’s problem is also in P.
  Weak Completion: NP‑hard (Theorem 13). Even without cycles, the follower’s freedom to ignore leader edges retains the NP‑hardness.

Thus, the “strong” variant is generally harder than the “weak” one, except in DAGs where the situation reverses.

Special Case: Constant Number of Leader Edges (|Eℓ| = O(1))

When the leader can influence only a constant number of edges, many variants become tractable (Table 2). However, for undirected graphs under strong completion the problem is shown to be equivalent to the well‑studied shortest‑k‑cycle problem (Theorem 18). The shortest‑k‑cycle problem asks for the minimum‑weight simple cycle that uses exactly k edges; its exact complexity is still open. This equivalence yields two notable consequences:

1. Randomized Polynomial‑time Algorithm – For polynomially bounded edge weights, the BSP problem admits a Monte‑Carlo algorithm running in polynomial time.
2. Derandomization Condition – The randomized algorithm can be derandomized (i.e., turned into a deterministic polynomial‑time algorithm) iff the shortest‑k‑cycle problem admits a deterministic polynomial‑time algorithm for constant k and polynomially bounded weights.

Thus, progress on the shortest‑k‑cycle problem directly translates into algorithmic advances for this restricted BSP variant.

Inapproximability

Most hardness results are strengthened to inapproximability statements (Section 6). Unless P = NP, none of the NP‑hard variants admit any polynomial‑time approximation algorithm with a bounded ratio. The Σ₂^P‑complete leader problem under strong completion is likewise not approximable within any reasonable factor unless the polynomial hierarchy collapses.

Relation to Existing Literature

The authors position the BSP problem within a broader class they term partitioned‑items bilevel problems: a ground set of items is split between leader and follower, each contributes items to a common combinatorial structure while optimizing separate objectives. They compare BSP to known bilevel problems such as toll‑setting (price‑setting) shortest‑path, inverse shortest‑path, and interdiction problems, noting that while those have been studied, none share the exact structure of simultaneously partitioned edge sets with distinct cost functions.

They also discuss connections to bilevel selection, knapsack, minimum spanning tree, and assignment problems, many of which are known to be Σ₂^P‑hard or NP‑hard. The BSP problem adds to this landscape by providing a natural graph‑theoretic example where Σ₂^P‑hardness can be proved using recent techniques from min‑max combinatorial optimization (citing