Complexity of the C"O"S-A- variant of Orbital Boundary Labeling

Determine the computational complexity of the ORBITAL BOUNDARY LABELING variant with free port candidates on the boundary circle (no pre-specified candidate ports), free cyclic order of labels along the boundary, and non-uniform label sizes under the port-ratio setting denoted A- in the COSA classification; specifically, ascertain whether this variant is NP-hard or admits a polynomial-time algorithm for minimizing the total leader length subject to non-crossing orbital-radial leaders.

Background

The paper introduces ORBITAL BOUNDARY LABELING for circular contours, categorizing problem variants by four dimensions (C, O, S, A) concerning candidate port positions, label order, label sizes, and port ratios. In Section 4.2.2 the authors analyze variants without candidate ports (free C) and with free label order (free O) when label sizes are non-uniform (S≠). They prove NP-hardness for several port-ratio settings (A categories) but leave one specific setting open.

The unresolved case is explicitly denoted as C"O"S-A-, referring to the combination of free candidate ports, free label order, non-uniform label sizes, and the A- port-ratio setting in their COSA classification. The open task is to establish whether this specific variant is NP-hard or polynomial-time solvable.