Subfunction Structure Matters: A New Perspective on Local Optima Networks
S. L. Thomson, M. W. Przewozniczek
TL;DR
This work addresses the gap where Local Optima Networks (LONs) are typically built and analyzed without leveraging problem structure. It introduces three LON construction approaches—Traditional black-box ILS, deterministic recombination with Partition Crossover (PX), and a gray-box approach (VIGP FIHCwLL)—and proposes subfunction-change metrics to capture how subfunction structure changes across edges. Across NK landscapes, MAX-SAT, and deceptive functions, the decomposition-aware methods reveal richer network structures, including denser connectivity and edge-wise subfunction transitions, offering new insights into optimization dynamics beyond classic LON metrics. The findings suggest that incorporating known or learnable subfunction structure into landscape analysis improves understanding of problem difficulty and search trajectories, with implications for linkage-learning optimizers and future scalability work; all data and code are available in a Zenodo repository.
Abstract
Local optima networks (LONs) capture fitness landscape information. They are typically constructed in a black-box manner; information about the problem structure is not utilised. This also applies to the analysis of LONs: knowledge about the problem, such as interaction between variables, is not considered. We challenge this status-quo with an alternative approach: we consider how LON analysis can be improved by incorporating subfunction-based information - this can either be known a-priori or learned during search. To this end, LONs are constructed for several benchmark pseudo-boolean problems using three approaches: firstly, the standard algorithm; a second algorithm which uses deterministic grey-box crossover; and a third algorithm which selects perturbations based on learned information about variable interactions. Metrics related to subfunction changes in a LON are proposed and compared with metrics from previous literature which capture other aspects of a LON. Incorporating problem structure in LON construction and analysing it can bring enriched insight into optimisation dynamics. Such information may be crucial to understanding the difficulty of solving a given problem with state-of-the-art linkage learning optimisers. In light of the results, we suggest incorporation of problem structure as an alternative paradigm in landscape analysis for problems with known or suspected subfunction structure.