  Subset selection is an interesting and important topic in the field of
evolutionary multi-objective optimization (EMO). Especially, in an EMO
algorithm with an unbounded external archive, subset selection is an essential
post-processing procedure to select a pre-specified number of solutions as the
final result. In this paper, we discuss the efficiency of greedy subset
selection for the hypervolume, IGD and IGD+ indicators. Greedy algorithms
usually efficiently handle subset selection. However, when a large number of
solutions are given (e.g., subset selection from tens of thousands of solutions
in an unbounded external archive), they often become time-consuming. Our idea
is to use the submodular property, which is known for the hypervolume
indicator, to improve their efficiency. First, we prove that the IGD and IGD+
indicators are also submodular. Next, based on the submodular property, we
propose an efficient greedy inclusion algorithm for each indicator. Then, we
demonstrate through computational experiments that the proposed algorithms are
much faster than the standard greedy subset selection algorithms.
