Efficient incremental non-dominated sorting for bi-objective problems via linear-space IDG representation

Abstract

This work presents an efficient incremental non-dominated sorting algorithm for bi-objective vectors. Non-dominated sorting algorithms organize solutions into layers, or Pareto fronts, based on dominance relations, and play a key role in the selection process of many well-known population-based algorithms for multi-objective optimization. Since the non-dominated sorting process is computationally expensive, its efficient design is of critical importance.

The proposed algorithm is incremental, that is, given a population whose solutions are already assigned to layers, a new solution is incorporated and the dominance structure is updated as efficiently as possible, reconstructing the layers with minimal computational effort. The method is based on irreducible domination graphs of order 1, IDG${}^1$, for which a storage structure requiring linear space in the number of solutions is introduced, instead of the quadratic requirements of the general case. Based on this representation, an incremental non-dominated sorting algorithm with $O\left(n\right)$ complexity is developed, matching the complexity of well-known algorithms in the literature while relying on simpler data structures. Key components of the approach include locating the appropriate front for the new solution, managing displaced nodes across layers, and efficiently updating dominance relations. This linear complexity is achieved thanks to the structure of the IDG${}^1$, which stores immediate dominance relations among solutions, i.e., the set of nodes in the previous Pareto layer that directly dominate each node.