Transitions from P to NP-hardness: the case of the Linear Ordering Problem

We decompose the linear ordering problem into a P and an NP-hard component by means of the Fourier transform. That is, we prove that the objective function can be expressed as the sum of two objective functions, one of which is associated with a P problem (an exact polynomial time algorithm is proposed to solve it), while the other is associated with an NP-hard problem. Based on this decomposition, we evaluate how different constructive algorithms whose behaviour only depends on univariate information degrade when the problem transits from P to NP-hard. A number of experiments are conducted with reduced dimensions, where the global optimum of the problems is known, giving different weights to the NP-hard component, while the weight of the P component is fixed.