Constant-factor approximation guarantee for GREEDY

Determine whether Algorithm GREEDY is a constant-factor approximation algorithm for the One-Sided Crossing Minimization problem.

Background

The paper defines Algorithm GREEDY to construct an ordering of the B-side vertices by processing unordered vertex pairs in non-decreasing order of the smaller of their two crossing numbers, orienting each pair according to the smaller crossing number whenever the relative order has not already been determined by transitivity. The authors ask whether the resulting ordering always achieves an objective value within a constant factor of the optimum for OSCM.

The question is connected to the subsequent cyclic crossing-number problem: the authors explain that a positive constant lower bound in that problem would imply that GREEDY is a constant-factor approximation algorithm. The paper proves partial bounds related to this connection but does not resolve the approximation question itself.