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

2025-07-30

AI Summary

This work addresses the computational inefficiency and analytical intractability of constructing Nearest-Better Networks (NBNs) for combinatorial optimization landscapes. We propose a theory-driven, logarithmic-linear-time NBN construction algorithm and establish, for the first time, a rigorous theoretical equivalence between the NBN and the maximum-probability transition network induced by a stochastic local search algorithm—thereby grounding landscape analysis in formal probability theory. Applying our method to OneMax and TSP benchmarks—and integrating behavioral data from EAX and LKH solvers—we uncover fundamental landscape properties: multimodality, neutrality, ruggedness, and deception. Specifically, OneMax exhibits multiscale structure; LKH is prone to deceptive local optima; and EAX suffers reduced cooperative search efficiency due to coexisting attraction basins. Our approach enables efficient, scalable NBN visualization and reveals intrinsic limitations of state-of-the-art TSP solvers under complex, high-dimensional fitness landscapes.

Technology Category

• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Computational social science

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.

Problem

• Extend NBN to combinatorial optimization problems efficiently
• Analyze algorithm behavior via NBN's maximum probability transition network
• Identify landscape features and algorithm limitations in OneMax and TSP

Innovation

• Efficient NBN computation with logarithmic linear time
• NBN as maximum probability transition network
• Applied NBN to OneMax and TSP problems