Exact and Fast Subset Selection Algorithms for the Bi-objective Integral R2 Indicator

Abstract

We study fixed-cardinality subset selection for the exact integral bi-objective $R_2$ indicator with a uniform continuum of weighted Tchebycheff scalarizing functions. The indicator measures the area under the lower envelope of scalarizing losses over weight space, rather than a finite sample average over weight vectors. For a sorted bi-objective Pareto-front approximation, represented by points ordered by increasing first objective and decreasing second objective, we derive an exact adjacent-neighbor decomposition of this integral objective into boundary terms, unary diagonal corrections, and selected-neighbor transition terms. This yields an exact Bellman dynamic program with $O(kn^2)$ running time for selecting $k$ of $n$ candidate points. We then prove that the transition matrix is Monge. This gives a divide-and-conquer implementation with $O(kn\log n)$ running time and, more strongly, a staircase matrix-search implementation with $O(kn)$ running time under constant-time arithmetic comparisons. The matrix-search proof is presented through a lower-envelope sweep over single-crossing transition functions and includes the triangular feasibility condition $i<j$. The algorithms are exact for the continuous integral $R_2$ setting and are distinct from finite-weight-vector approximations, although they are related to earlier exact and dynamic-programming work on two-dimensional indicator-based subset selection, including hypervolume subset selection. Reproducible Python code compares exhaustive enumeration, the direct left-to-right dynamic program, the divide-and-conquer dynamic program, and the matrix-search implementation under explicit consistency checks.