  The hypervolume indicator is one of the most used set-quality indicators for
the assessment of stochastic multiobjective optimizers, as well as for
selection in evolutionary multiobjective optimization algorithms. Its
theoretical properties justify its wide acceptance, particularly the strict
monotonicity with respect to set dominance which is still unique of
hypervolume-based indicators. This paper discusses the computation of
hypervolume-related problems, highlighting the relations between them,
providing an overview of the paradigms and techniques used, a description of
the main algorithms for each problem, and a rundown of the fastest algorithms
regarding asymptotic complexity and runtime. By providing a complete overview
of the computational problems associated to the hypervolume indicator, this
paper serves as the starting point for the development of new algorithms, and
supports users in the identification of the most appropriate implementations
available for each problem.
