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  "path": "/abs/2606.23365v1",
  "publishedAt": "2026-06-23T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Michael T. M. Emmerich"
  ],
  "textContent": "**Authors:** Michael T. M. Emmerich\n\nWe 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",
  "title": "Exact and Fast Subset Selection Algorithms for the Bi-objective Integral R2 Indicator"
}