Complexity of maximizing forward arcs in Hamilton cycles of dense oriented graphs

Determine whether finding a Hamilton oriented cycle with the maximum possible number of forward arcs is NP-hard for oriented graphs on n ≥ 3 vertices satisfying d(u) ≥ n/2 for every vertex u, and for the broader class of oriented graphs satisfying the Ore-type condition in Conjecture 1.3.

Background

The paper develops polynomial-time algorithms for maximizing the number of forward arcs in Hamilton oriented paths and cycles for semicomplete multipartite digraphs and locally semicomplete digraphs. It then shows that analogous problems can be NP-hard in broader classes, including locally out-semicomplete and quasi-transitive digraphs, despite polynomial-time solvability of some Hamilton-cycle decision problems.

The authors explicitly leave unresolved whether the optimization problem is NP-hard even in the dense oriented-graph class covered by the known minimum-degree theorem, and whether the same hardness holds for the class defined by Conjecture 1.3.