Visualizing Multimodality in Combinatorial Search Landscapes

TL;DR

This paper investigates how to visualize multimodality in combinatorial search landscapes. It surveys several visualization paradigms—Distance--Fitness correlation, Local Optima Networks, Hinged Bitstring Maps, Sequence Index Plots, Search Trajectory Networks, and Violation Landscapes—and frames them within the Grammar of Graphics to enable principled comparison and composition. It then proposes a simple framework for combining views through juxtaposition and superimposition, illustrating benefits and limitations with case studies. The work highlights that no single visualization captures all structure, and it points to future work such as animated aesthetics and attractor-network approaches to enrich multimodal landscape analysis.

Abstract

This work walks through different visualization techniques for combinatorial search landscapes, focusing on multimodality. We discuss different techniques from the landscape analysis literature, and how they can be combined to provide a more comprehensive view of the search landscape. We also include examples and discuss relevant work to show how others have used these techniques in practice, based on the geometric and aesthetic elements of the Grammar of Graphics. We conclude that there is no free lunch in visualization, and provide recommendations for future work as there are several paths to continue the work in this field.

Figures (9)

• Figure 1: The landscapes of some 2D test functions in the continuous domain vanaretCertifiedGlobalMinima2020. The $z$-axis is used to plot the fitness, while the $x$- and $y$-axes are function parameters (shown here as $x_1$ and $x_2$).
• Figure 2: Analysis of 2511 local optima, on the feature selection problem of the Credit Approval credit_approval_27 dataset using a decision tree classifier. The $y$-axis shows the quality of a solution (accuracy of classification) while the $x$-axis shows average Hamming distance between local optima (cf. left panel), and Hamming distance from each local optima to its closest global optimum (cf. right panel).
• Figure 3: Hex-bin plot of distance correlation for the Heart Disease (Cleveland) dataset heart_disease_45, using a decision tree classifier under four different levels of regularization. Each bin aggregates different number of local optima, $\boldsymbol{b}^+$. A darker shade means a higher concentration of $\boldsymbol{b}^+$.
• Figure 4: An STN comparing two algorithms: Biased Random-Key Genetic Algorithm and Ant Colony Optimization using the discrete example problem from Ochoa et al. ochoaSearchTrajectoryNetworks2021. The plot was generated using the STNs online tool.
• Figure 5: Fitness and violation landscapes of a 2D Constrained Ackley Function. The constraint is handled as a penalty for all solutions $\boldsymbol{b}$ with $f(\boldsymbol{b})>15$, creating two regions in $\mathcal{X}$.