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

TL;DR

This paper investigates how the performance of constructive heuristics deteriorates as the Linear Ordering Problem transitions from polynomial-time solvable to NP-hard, by analyzing the impact of its decomposed objective functions.

Contribution

It introduces a decomposition of the Linear Ordering Problem's objective into P and NP-hard components and studies heuristic performance relative to their contributions.

Findings

Heuristic performance declines with increased NP-hard component weight.

The objective function can be split into a P and an NP-hard part.

Experimental results show degradation in solution quality as NP-hard influence grows.

Abstract

In this paper we evaluate how constructive heuristics degrade when a problem transits from P to NP-hard. This is done by means of the linear ordering problem. More specifically, for this problem 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. We study how different constructive algorithms whose behaviour only depends on univariate information perform depending on the contribution of the P or NP-hard components of the problem. 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. It is observed how the performance of the constructive algorithms gets worse as the weight given to the NP-hard component increases.