# Novel algorithm to calculate hypervolume indicator of Pareto approximation set

## TL;DR

This paper introduces a recursive vertex-splitting algorithm for calculating the hypervolume indicator of Pareto approximation sets in multi-objective optimization, aiming to improve efficiency in higher dimensions.

## Contribution

It presents a novel recursive algorithm with a strategic splitting method and analyzes its complexity, offering an alternative to existing methods for hypervolume calculation.

## Key findings

- Achieves worst-case time complexity of O((d/2)^n)
- Uses a carefully chosen splitting reference point to reduce computation
- Provides a new approach for hypervolume calculation in higher dimensions

## Abstract

Hypervolume indicator is a commonly accepted quality measure for comparing Pareto approximation set generated by multi-objective optimizers. The best known algorithm to calculate it for $n$ points in $d$-dimensional space has a run time of $O(n^{d/2})$ with special data structures. This paper presents a recursive, vertex-splitting algorithm for calculating the hypervolume indicator of a set of $n$ non-comparable points in $d>2$ dimensions. It splits out multiple child hyper-cuboids which can not be dominated by a splitting reference point. In special, the splitting reference point is carefully chosen to minimize the number of points in the child hyper-cuboids. The complexity analysis shows that the proposed algorithm achieves $O((\frac{d}{2})^n)$ time and $O(dn^2)$ space complexity in the worst case.