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

Abstract: 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.

• The paper characterizes maximum forward arcs in semicomplete multipartite digraphs using maximum-cost path-cycle and cycle factors, enabling polynomial-time solutions through minimum-cost flow.
• The paper proves that locally semicomplete digraphs have cycle optimum n in the strong case, n minus the end-component distance when non-strong and 2-connected, and no Hamilton cycle otherwise.
• The paper shows tractability is class-specific: despite easy Hamiltonicity tests, forward-arc maximization remains NP-hard for locally out-semicomplete and quasi-transitive digraphs.

Background and motivation

The study of oriented discrepancy in digraphs originates in Erdős's 1963 notion of discrepancy for 2-edge-colored undirected graphs. Gishboliner, Krivelevich, and Michaeli introduced the directed analog: for a digraph $D$ possessing a Hamilton oriented cycle, the directed discrepancy is the maximum of $\sigma_{\max}(C)$ over all Hamilton oriented cycles $C$, where $\sigma^+(C)$ and $\sigma^-(C)$ count forward and backward arcs respectively (2602.10713). Since $$\sigma_{\max}(C) = \max\{\sigma^+(C), \sigma^+(C^{-1})\}$$, maximizing $\sigma_{\max}$ reduces to maximizing $\sigma^+$, so the authors work directly with the Maximum-Forward-Arc Hamilton Oriented Cycle/Path problem (MFAHOC/MFAHOP): given a digraph $D$, decide whether $D$ contains a Hamilton oriented cycle (path) and, if so, find one with the greatest number of forward arcs.

The driving conjecture was a Dirac-type generalization by Gishboliner et al., proved by Freschi and Lo: every $\sigma_{\max}(C)$0-vertex oriented graph with minimum degree $\sigma_{\max}(C)$1 has a Hamilton oriented cycle with at least $\sigma_{\max}(C)$2 forward arcs. The present paper takes an algorithmic and structural approach to MFAHOC/MFAHOP on generalizations of tournaments — semicomplete multipartite digraphs and locally semicomplete digraphs — obtaining exact characterizations of $\sigma_{\max}(C)$3 and $\sigma_{\max}(C)$4 together with polynomial-time algorithms.

A key hardness observation frames the scope of these results: MFAHOC is NP-hard even for oriented graphs with $\sigma_{\max}(C)$5. The reduction joins an arbitrary $\sigma_{\max}(C)$6-vertex oriented graph $\sigma_{\max}(C)$7 to a vertex-disjoint tournament $\sigma_{\max}(C)$8 on $\sigma_{\max}(C)$9 vertices via all arcs from $C$0 to $C$1; the resulting digraph has a Hamilton oriented cycle with $C$2 forward arcs if and only if $C$3 has a Hamilton path. Thus high minimum degree guarantees existence but does not make optimization tractable, which motivates restricting attention to structured classes where Hamiltonicity itself is polynomial-time decidable.

Semicomplete multipartite digraphs

A semicomplete multipartite digraph arises from a complete multipartite graph by orienting each edge in one or both directions; its maximal independent sets are the partite sets, with sizes $C$4. Existence of Hamilton oriented paths and cycles depends only on the partite set sizes via simple majority inequalities: the HC-majority inequality $C$5 characterizes Hamilton oriented cycles, and the HP-majority inequality $C$6 characterizes Hamilton oriented paths. These follow by reducing to Hamiltonicity of the complete multipartite host graph and applying Dirac's theorem.

The central tool is the symmetric $C$7-digraph $C$8: each arc of $C$9 receives cost 1, and for every arc $\sigma^+(C)$0 with $\sigma^+(C)$1, a reverse arc $\sigma^+(C)$2 of cost 0 is added. Under this construction, $\sigma^+(C)$3 equals the maximum cost of a Hamilton path in $\sigma^+(C)$4, and similarly for cycles. The paper's first main result identifies this quantity with a flow-computable parameter:

Object	Characterization
$\sigma^+(C)$5	equals $\sigma^+(C)$6, the max cost of a 1-path-cycle factor in $\sigma^+(C)$7
$\sigma^+(C)$8	equals $\sigma^+(C)$9, the max cost of a cycle factor in $\sigma^-(C)$0, unless $\sigma^-(C)$1 and $\sigma^-(C)$2 is not hamiltonian, in which case $\sigma^-(C)$3

For paths, the proof exploits Gutin's characterization of hamiltonian-path semicomplete multipartite digraphs via 1-path-cycle factors: given a maximum-cost factor $\sigma^-(C)$4, the spanning subdigraph $\sigma^-(C)$5 with arc set $\sigma^-(C)$6 has a Hamilton path whose zero-cost arcs are no more numerous than those of $\sigma^-(C)$7, forcing equality. A maximum-cost 1-path-cycle factor is obtained in polynomial time via a minimum-cost flow formulation on an auxiliary network with source $\sigma^-(C)$8, sink $\sigma^-(C)$9, and unit lower/upper bounds on all vertices.

For cycles, the argument splits into two cases. If $$\sigma_{\max}(C) = \max\{\sigma^+(C), \sigma^+(C^{-1})\}$$0, some factor arc has cost 0; deleting it yields a 1-path-cycle factor, and a new structural lemma — if a 1-path-cycle factor has a path whose endpoints lie in different partite sets, then $$\sigma_{\max}(C) = \max\{\sigma^+(C), \sigma^+(C^{-1})\}$$1 has a Hamilton path with endpoints in different partite sets — produces a Hamilton oriented cycle whose cost matches that of $$\sigma_{\max}(C) = \max\{\sigma^+(C), \sigma^+(C^{-1})\}$$2. This lemma is proved using Yeo's theorem on weakly-dominating ordered cycle factors: any non-Hamilton cycle factor can be refined either to a Hamilton cycle or to a factor whose cycles admit an ordering under weak domination, which permits sequential merging of cycles while controlling endpoint partite membership. If instead $$\sigma_{\max}(C) = \max\{\sigma^+(C), \sigma^+(C^{-1})\}$$3 and $$\sigma_{\max}(C) = \max\{\sigma^+(C), \sigma^+(C^{-1})\}$$4 is not hamiltonian, removing one arc from the factor and applying the same lemma gives a Hamilton oriented cycle with exactly one backward arc, hence $$\sigma_{\max}(C) = \max\{\sigma^+(C), \sigma^+(C^{-1})\}$$5.

Two implications deserve emphasis. First, both optima are computable in polynomial time, extending the classical polynomial-time results of Bang-Jensen, Gutin, and Yeo for the Hamilton cycle decision problem — notably, no structural characterization of hamiltonian semicomplete multipartite digraphs is known, yet the forward-arc optimum is fully characterized regardless. Second, the characterization is exact rather than approximate: when a Hamilton oriented cycle exists, the gap between the best achievable value and $$\sigma_{\max}(C) = \max\{\sigma^+(C), \sigma^+(C^{-1})\}$$6 is precisely determined by whether the maximum-cost cycle factor already spans as a single cycle.

Locally semicomplete digraphs

A digraph is locally semicomplete if the out-neighbors and in-neighbors of every vertex induce semicomplete subdigraphs. Every connected locally semicomplete digraph has a Hamilton path, so MFAHOP is trivial here; the interest lies in cycles. For strong locally semicomplete digraphs, Bang-Jensen's theorem guarantees a Hamilton (directed) cycle, so $$\sigma_{\max}(C) = \max\{\sigma^+(C), \sigma^+(C^{-1})\}$$7. The substantive case is non-strong digraphs whose underlying graph is 2-connected.

The characterization rests on the unique acyclic ordering $$\sigma_{\max}(C) = \max\{\sigma^+(C), \sigma^+(C^{-1})\}$$8 of strong components, which satisfies strong dominance properties: consecutive components dominate one another, and any arc from $$\sigma_{\max}(C) = \max\{\sigma^+(C), \sigma^+(C^{-1})\}$$9 to $\sigma_{\max}$0 forces domination across all intermediate components. Two complementary bounds pin down the optimum:

• Upper bound construction: when $\sigma_{\max}$1 is 2-connected, a greedy shortest $\sigma_{\max}$2-path $\sigma_{\max}$3 can be chosen so that deleting its internal vertices leaves a connected digraph (the 2-connectivity rules out cut vertices, and the component ordering then forces domination across the deleted components). A Hamilton path through the remainder, running from $\sigma_{\max}$4 to $\sigma_{\max}$5, closes with $\sigma_{\max}$6 traversed backward to form a Hamilton oriented cycle $\sigma_{\max}$7 whose only backward arcs lie on $\sigma_{\max}$8, giving $\sigma_{\max}$9.
• Matching lower bound: every oriented $\sigma^+$0-path contains at least $\sigma^+$1 forward arcs, proved by extracting a monotone subsequence of vertices with strictly increasing component indices; the corresponding arcs must be forward by the component-ordering properties, and they form a $\sigma^+$2-path no shorter than the distance. Any Hamilton oriented cycle must traverse a $\sigma^+$3 segment whose reversal is such a path, so it incurs at least $\sigma^+$4 backward arcs.

Consequently, for connected non-strong locally semicomplete $\sigma^+$5 with $\sigma^+$6: $\sigma^+$7 if $\sigma^+$8 is 2-connected, and $\sigma^+$9 otherwise (a Hamilton oriented cycle would survive deletion of any single vertex, contradicting non-2-connectivity). All cases are solvable in polynomial time, since acyclic orderings, shortest paths, and Hamilton paths/cycles in these classes are all polynomial-time computable.

Complexity beyond these classes

The paper explicitly refutes the natural conjecture that polynomial-time solvability of the Hamilton cycle problem implies tractability of the forward-arc maximization variant within a class of tournament generalizations. Two counterexamples are given:

• Locally out-semicomplete digraphs (only out-neighborhoods need be semicomplete): Hamiltonicity is equivalent to strong connectivity, hence polynomial-time decidable. Yet the digraph obtained from a balanced bipartite graph $D$0 by orienting all edges from $D$1 to $D$2 and completing $D$3 arbitrarily is locally out-semicomplete and has a Hamilton oriented cycle iff $D$4 has a Hamilton cycle — NP-hard.
• Quasi-transitive digraphs: Hamiltonicity is polynomial-time decidable, but the same bipartite construction yields a quasi-transitive (indeed transitive, acyclic, bipartite) digraph whose Hamilton oriented cycle existence is NP-hard.

These examples show that the positive results depend essentially on the bidirectional local completeness of the two classes studied, not merely on their Hamiltonicity being easy.

Limitations and open questions

The characterizations apply only to semicomplete multipartite and locally semicomplete digraphs; the paper demonstrates that extending them to other generalizations of tournaments (locally out-semicomplete, quasi-transitive) is impossible unless P = NP, since even the decision version becomes hard there. Within the treated classes, the results presuppose the majority inequalities for existence and, for the cycle case, rely on Yeo's weak-domination structure; the absence of a full structural characterization of hamiltonian semicomplete multipartite digraphs remains open, though it is not needed for the algorithmic results. Whether analogous exact characterizations exist for intermediate classes between locally out-semicomplete and locally semicomplete digraphs is not addressed.

Conclusion

This paper settles MFAHOC and MFAHOP exactly for two standard generalizations of tournaments. In semicomplete multipartite digraphs, the forward-arc optima coincide with maximum-cost 1-path-cycle and cycle factors in the symmetric $D$5-digraph, computable via minimum-cost flows; in locally semicomplete digraphs, the cycle optimum is $D$6 in the 2-connected non-strong case, $D$7 in the strong case, and nonexistent otherwise. Both characterizations yield polynomial-time algorithms, and the accompanying hardness constructions delineate precisely where such tractability ends.