Nearest-Better Network for Visualizing and Analyzing Combinatorial Optimization Problems: A Unified Tool

TL;DR

This paper introduces an efficient method for constructing Nearest-Better Networks to analyze combinatorial optimization landscapes, revealing key features and limitations of algorithms like EAX and LKH on TSP and OneMax problems.

Contribution

It provides a theoretical foundation for NBN as a maximum probability transition network and introduces a logarithmic time complexity algorithm for NBN computation.

Findings

OneMax landscape shows neutrality, ruggedness, and modality.

TSP challenges include ruggedness, modality, and deception.

EAX and LKH algorithms have limitations in handling TSP landscape features.

Abstract

The Nearest-Better Network (NBN) is a powerful method to visualize sampled data for continuous optimization problems while preserving multiple landscape features. However, the calculation of NBN is very time-consuming, and the extension of the method to combinatorial optimization problems is challenging but very important for analyzing the algorithm's behavior. This paper provides a straightforward theoretical derivation showing that the NBN network essentially functions as the maximum probability transition network for algorithms. This paper also presents an efficient NBN computation method with logarithmic linear time complexity to address the time-consuming issue. By applying this efficient NBN algorithm to the OneMax problem and the Traveling Salesman Problem (TSP), we have made several remarkable discoveries for the first time: The fitness landscape of OneMax exhibits neutrality, ruggedness, and modality features. The primary challenges of TSP problems are ruggedness, modality, and deception. Two state-of-the-art TSP algorithms (i.e., EAX and LKH) have limitations when addressing challenges related to modality and deception, respectively. LKH, based on local search operators, fails when there are deceptive solutions near global optima. EAX, which is based on a single population, can efficiently maintain diversity. However, when multiple attraction basins exist, EAX retains individuals within multiple basins simultaneously, reducing inter-basin interaction efficiency and leading to algorithm's stagnation.