Polynomial-time algorithm for longest prescribed-endpoint paths in semicomplete digraphs

Establish whether a longest $(x,y)$-path in an arbitrary semicomplete digraph can be found in polynomial time for prescribed distinct vertices $x$ and $y$.

Background

The paper studies the optimization problem of finding a maximum-order directed path whose first and last vertices are prescribed. A polynomial-time algorithm is known for deciding and constructing a Hamiltonian (x,y)(x,y)-path when one exists, but that algorithm does not optimize the number of vertices used by the path.

The paper proves polynomial-time solvability when the endpoints lie in different strong components and gives an exact enumeration algorithm that is polynomial for every fixed number of omitted vertices. The general case, particularly when the endpoints lie in the same strong component and the number of omitted vertices is unrestricted, remains unresolved.

References

Bang-Jensen and Gutin therefore posed the following open conjecture . There is a polynomial-time algorithm that, given a semicomplete digraph $D$ and distinct vertices $x,y\in V(D)$, finds a longest $(x,y)$-path.

— Paths with Prescribed Endpoints in Semicomplete and Locally Semicomplete Digraphs  (2608.17439 - Bai et al., 18 Aug 2026) in Conjecture 1 in Section 1; restated as Problem 1 in Section 5