Binary shape-based benchmarks for analyzing search preferences in multi-objective combinatorial optimization

Existing multi-objective benchmarks for combinatorial optimization are mainly designed to evaluate convergence and diversity in the objective space, but they offer limited insight into how algorithms explore discrete search spaces under binary representations. In particular, when different algorithms achieve similar Pareto front approximations, it remains unclear whether they exploit the same regions of the decision space or rely on fundamentally different binary structures. To address this limitation, we propose a family of binary shape-based benchmark problems that explicitly decouple three key aspects: (1) the geometry of the Pareto-optimal region, (2) the mapping from binary decision vectors to a low-dimensional geometric space, and (3) analysis methods for revealing search preferences. The proposed benchmarks construct multiple distance-based objectives with respect to simple geometric shapes, including circles, triangles, rectangles, and regular k-gons. Each shape induces a Pareto-optimal region whose interior, edges, and corners correspond to distinct trade-off patterns. In addition, we introduce a cluster-based genotypic analysis framework. Solutions are grouped in the Hamming space of binary decision vectors, characteristic bit-frequency profiles are extracted for each cluster, and the resulting clusters are visualized in the geometric space. Experimental studies using different types of multi-objective evolutionary and local search algorithms demonstrate that methods with similar objective-space performance can exhibit markedly different preferences for regions of the Pareto-optimal set and for genotypic clusters.