On the Effect of Connectedness for Biobjective Multiple and Long Path Problems

Recently, the property of connectedness has been claimed to give a strong motivation on the design of local search techniques for multiobjective combinatorial optimization (MOCO). Indeed, when connectedness holds, a basic Pareto local search, initialized with at least one non-dominated solution, allows to identify the efficient set exhaustively. However, this becomes quickly infeasible in practice as the number of efficient solutions typically grows exponentially with the instance size. As a consequence, we generally have to deal with a limited-size approximation, where a good sample set has to be found. In this paper, we propose the biobjective multiple and long path problems to show experimentally that, on the first problems, even if the efficient set is connected, a local search may be outperformed by a simple evolutionary algorithm in the sampling of the efficient set. At the opposite, on the second problems, a local search algorithm may successfully approximate a disconnected efficient set. Then, we argue that connectedness is not the single property to study for the design of local search heuristics for MOCO. This work opens new discussions on a proper definition of the multiobjective fitness landscape.

💡 Research Summary

The paper investigates the role of connectedness—a property indicating that the set of Pareto‑optimal (efficient) solutions forms a single connected component under a chosen neighborhood—in guiding the design of local‑search heuristics for multi‑objective combinatorial optimization (MOCO). While prior work has argued that, if the efficient set is connected, a basic Pareto Local Search (PLS) initialized with at least one non‑dominated solution can, in principle, enumerate the entire efficient set, this guarantee quickly becomes impractical because the number of efficient solutions typically grows exponentially with problem size. Consequently, practitioners must settle for a limited‑size approximation of the efficient set, and the central question becomes how to obtain a high‑quality sample of that set under realistic computational budgets.

To explore the practical implications of connectedness, the authors introduce two synthetic benchmark problems, each designed to stress different aspects of the search process:

1. Bi‑objective Multiple Path (BMP) – The efficient set is mathematically connected, but the distance (in Hamming space) between successive efficient solutions is deliberately large, and many local optima exist that are not efficient. This construction makes it difficult for a purely local search that only examines immediate neighbors to progress beyond the initial region.

2. Bi‑objective Long Path (BLP) – The efficient set is deliberately disconnected, consisting of several isolated components. Within each component, however, the efficient solutions lie on a long, monotone “path” such that moving by a single bit flip always stays inside the component. Thus, a local search can traverse an entire component once it reaches it.

The experimental framework compares a standard PLS algorithm with a very simple evolutionary algorithm (EA). The EA uses a fixed population, uniform 1‑bit mutation, and a selection scheme based on Pareto dominance and crowding distance. Both algorithms are given the same evaluation budget (number of objective function calls) and are evaluated using hyper‑volume, additive ε‑indicator, and coverage ratio of the true efficient set.

Results on BMP: PLS quickly stalls because its neighborhood exploration rarely discovers new non‑dominated solutions; the algorithm spends most of its budget evaluating dominated neighbors. In contrast, the EA’s stochastic mutations allow it to “jump” across large Hamming distances, thereby sampling a much broader region of the search space. Even with modest evaluation budgets, the EA attains hyper‑volume values 2–3 times larger than PLS and covers a substantially higher proportion of the efficient set.

Results on BLP: The situation reverses. Because each component’s efficient solutions are linked by single‑bit moves, PLS can systematically walk along the long path once it lands in a component, achieving high coverage with relatively few evaluations. The EA, however, suffers from its random walk nature; it frequently mutates out of a component before fully traversing it, leading to lower hyper‑volume and ε‑indicator scores compared with PLS, especially under tight evaluation limits.

These complementary findings demonstrate that connectedness alone is insufficient to predict the success of a local‑search method. A connected efficient set can still be “hard” for local search if the landscape is rugged, the distances between efficient solutions are large, or many non‑efficient local optima act as traps. Conversely, a disconnected efficient set can be easy to approximate if each component possesses a simple internal structure that aligns with the neighborhood definition.

The authors argue that the notion of a multi‑objective fitness landscape must be enriched beyond connectedness. Features such as ruggedness, modality, distribution of distances between efficient solutions, and the alignment between the neighborhood operator and the geometry of the efficient set should be considered when designing or selecting heuristics. Moreover, hybrid strategies that combine global stochastic moves (as in evolutionary algorithms) with focused local refinement (as in PLS) may offer a more robust approach across diverse problem classes.

In conclusion, while connectedness provides a useful theoretical lens, it should not be the sole guiding principle for MOCO algorithm design. Future work is encouraged to develop quantitative metrics that capture the broader landscape characteristics and to explore adaptive algorithms that can detect and exploit these properties on‑the‑fly. This line of research promises to yield more reliable and efficient methods for approximating Pareto fronts in real‑world combinatorial problems.