P-NP Instance Decomposition Based on the Fourier Transform for Solving the Linear Ordering Problem
提出一种基于傅里叶变换的实例分解策略，将线性排序问题分解为P和NP难两部分，通过调整比例构造人工实例，并设计P-Descent Search算法逐步优化，实验表明该方法性能可预测，有助于自动调参。
The Fourier transform over finite groups has proved to be a useful tool for analyzing combinatorial optimization problems. However, few heuristic and metaheuristic algorithms have been proposed in the literature that utilize the information provided by this technique to guide the search process. In this work, we attempt to address this research gap by considering the case study of the Linear Ordering Problem (LOP). Based on the Fourier transform, we propose an instance decomposition strategy that divides any LOP instance into the sum of two LOP instances associated with a P and an NP-Hard optimization problem. By linearly aggregating the instances obtained from the decomposition, it is possible to create artificial instances with modified proportions of the P and NP-Hard components. Conducted experiments show that increasing the weight of the P component leads to a less rugged fitness landscape suitable for local search-based optimization. We take advantage of this phenomenon by presenting a new metaheuristic algorithm called P-Descent Search (PDS). The proposed method, first, optimizes a surrogate instance with a high proportion of the P component, and then, gradually increases the weight of the NP-Hard component until the original instance is reached. The multi-start version of PDS shows a promising and predictable performance that appears to be correlated to specific characteristics of the problem, which could open the door to an automatic tuning of its hyperparameters.