Forward Arc Maximization for Hamilton Oriented Cycles and Paths in Generalizations of Tournaments

Gishboliner, Krivelevich, and Michaeli (2023) conjectured the following generalization of Dirac’s theorem: If the minimum degree $δ$ of an $n$-vertex oriented graph $G$ is greater or equal to $n/2$, then $G$ has a Hamilton oriented cycle with at least $δ$ forward arcs. Freschi and Lo (2024) proved this conjecture. In this paper, we study the problem of maximizing the number of forward arcs in Hamilton oriented cycles/paths in generalizations of tournaments. We obtain characterizations for the maximum number of forward arcs in semicomplete multipartite digraphs and locally semicomplete digraphs. These characterizations lead to polynomial-time algorithms. Note that the above problems are NP-hard for some other generalizations of tournaments even though the Hamilton cycle problem is polynomial-time solvable for these digraph classes.

💡 Research Summary

The paper investigates the problem of maximizing the number of forward arcs in Hamiltonian oriented cycles and paths, a natural optimization extension of the recent result by Freschi and Lo (2024) that confirmed the conjecture of Gishboliner, Krivelevich, and Michaeli (2023). The conjecture, a directed analogue of Dirac’s theorem, states that any n‑vertex oriented graph with minimum degree δ ≥ n/2 contains a Hamiltonian oriented cycle with at least δ forward arcs. While the existence of such a cycle is now known, the authors ask for the maximum possible number of forward arcs and study the computational complexity of this optimization problem, which they denote MF‑AHOC (Maximum‑Forward‑Arc Hamiltonian Oriented Cycle) and its path analogue MF‑AHOP.

The authors first show that MF‑AHOC/MF‑AHOP is NP‑hard even for oriented graphs with δ ≥ n/2, by a reduction that embeds a Hamiltonian path problem into a forward‑arc counting problem. Consequently, they focus on two well‑structured generalizations of tournaments where Hamiltonicity is already polynomial‑time decidable: semicomplete multipartite digraphs and locally semicomplete digraphs.

Semicomplete multipartite digraphs

A semicomplete multipartite digraph is obtained from a complete p‑partite graph by replacing each inter‑partite edge with either a single directed arc, its reverse, or a pair of opposite arcs. Let the partite set sizes be n₁,…,n_p. The authors introduce two simple “majority” inequalities:

• HC‑majority: 2·max_i n_i ≤ ∑_i n_i (necessary and sufficient for a Hamiltonian cycle).
• HP‑majority: 2·max_i n_i ≤ ∑_i n_i + 1 (necessary and sufficient for a Hamiltonian path).

These inequalities are direct extensions of Dirac’s condition to the multipartite setting.

To translate forward‑arc maximization into a combinatorial optimization problem, they construct the symmetric (0,1)‑digraph (\widehat D): each original arc receives cost 1, and a missing opposite arc (if it does not already exist) is added with cost 0. For any oriented path or cycle, the number of forward arcs equals the total cost of the corresponding walk in (\widehat D).

Theorem 1.3 (paths). If D satisfies HP‑majority, then the maximum number of forward arcs over all Hamiltonian paths equals the maximum cost of a 1‑path‑cycle factor in (\widehat D) (a spanning subgraph consisting of a single path together with vertex‑disjoint cycles). The authors show that this factor can be obtained via a minimum‑cost flow computation on a transformed network, yielding a polynomial‑time algorithm that simultaneously produces the optimal forward‑arc count and an explicit Hamiltonian path achieving it.

Theorem 1.4 (cycles). If D satisfies HC‑majority, the analogous statement holds with cycle factors (spanning collections of cycles). The optimal forward‑arc count is the maximum cost of a cycle factor, except when that cost equals n while D is not Hamiltonian; in that exceptional case the optimum is n − 1. Again, a min‑cost flow reduction provides a polynomial‑time solution.

A supporting structural result (Theorem 1.5) shows that if a 1‑path‑cycle factor has its path endpoints in distinct partite sets, the whole digraph contains a Hamiltonian path whose endpoints also lie in distinct partite sets. This leverages a theorem of Yeo on weak domination among cycles.

Overall, for semicomplete multipartite digraphs the forward‑arc maximization problem is completely characterized and solvable in polynomial time, despite the underlying NP‑hardness in general oriented graphs.

Locally semicomplete digraphs

A digraph D is locally semicomplete if for every vertex x, the subdigraph induced by its out‑neighbourhood N⁺(x) and the subdigraph induced by its in‑neighbourhood N⁻(x) are both semicomplete. Such digraphs are known to always possess a Hamiltonian path when the underlying undirected graph is connected, and every strongly connected locally semicomplete digraph has a Hamiltonian cycle.

The authors analyze the structure of the strong components of a non‑strongly connected locally semicomplete digraph. These components admit a unique acyclic ordering (C_1, C_2, …, C_\ell) such that there are no arcs from a later component to an earlier one. The distance (d(C_1, C_\ell)) in the underlying undirected graph (i.e., the length of a shortest undirected path between the first and last components) becomes the key parameter.

Theorem 1.6 (optimal forward arcs).

• If D is strongly connected, every Hamiltonian cycle can be oriented so that all n arcs are forward, giving σ⁺ₘₐₓ = n.
• If D is not strongly connected but its underlying graph is 2‑connected, then σ⁺ₘₐₓ = n − d(C₁, C_ℓ). The subtraction accounts for the unavoidable backward arcs needed to traverse from the first to the last strong component.
• If the underlying graph is not 2‑connected, D has no Hamiltonian cycle, so σ⁺ₘₐₓ = 0.

The proof constructs an optimal cycle by first linking the strong components along a shortest undirected path and then orienting each component internally as a directed cycle. The distance term precisely counts the minimal number of backward arcs forced by the component layout. All steps—identifying strong components, computing the acyclic order, and finding the shortest inter‑component path—are doable in polynomial time, yielding an explicit optimal Hamiltonian cycle when one exists.

Complexity landscape and conclusions

The paper demonstrates that, while the decision version of Hamiltonicity is polynomial for the two classes considered, the optimization version (maximizing forward arcs) can be either polynomial (semicomplete multipartite, locally semicomplete) or NP‑hard (other tournament generalizations). This dichotomy highlights that adding a natural quantitative objective can dramatically change computational difficulty.

Methodologically, the work blends classic graph‑theoretic characterizations (majority inequalities, strong component ordering) with algorithmic tools from combinatorial optimization (minimum‑cost flow, network transformations). The symmetric (0,1)‑digraph construction provides a clean bridge between forward‑arc counting and weighted path/cycle problems, enabling the use of well‑studied polynomial algorithms.

In summary, the authors provide a complete structural and algorithmic treatment of forward‑arc maximization for Hamiltonian cycles and paths in two important tournament‑like digraph families, extending the recent Dirac‑type existence result to a full optimization framework and delineating the boundary between tractable and intractable cases.