{
  "abstract": "Suppose an optimization problem is randomly restricted, independently on each variable, and re-optimized after the restriction is revealed. For multilinear objectives, every fixed solution is unbiased after rescaling, but re-optimizing introduces an upward bias. We bound this excess, the hindsight gain, over products of arbitrary compact local domains in Euclidean unit balls. For interactions $A=(A_e)_e$ among $r$ distinct variables and independent random weights $ζ_i$, the gain is at most $O_r(\\|A\\|_F M^{(r-1)/2}σ_{\\max})$, where $\\|A\\|_F^2$ is the sum of squared coefficients, $M=\\sum_i\\mathbb{E}ζ_i^2$, and $σ_{\\max}^2=\\max_i\\mathrm{Var}(ζ_i)$; only second moments are needed. Under uniform retention with probability $p$, we sharpen this to $O_r(p^{r-1/2}\\sqrt{1-p}\\,n^{(r-1)/2}\\|A\\|_F)$, which is tight for every fixed $r$ and $p\\in(0,1)$. Our proof combines a comparison principle along a single support direction with tensor projection, vector contraction, and repeated thinning. Applications: (i) For fixed arity and alphabet size, solving the subproblem induced by $O(\\varepsilon^{-2})$ sampled variables estimates a dense Max-$r$CSP optimum within $\\pm\\varepsilon n^r$, and this sample size is optimal. (ii) The same approach estimates MaxCut and the distance to bipartiteness within $\\pm\\varepsilon n^2$ using $O(\\varepsilon^{-4})$ nonadaptive queries and $O(\\varepsilon^{-2})$ examined vertices, both optimal. (iii) We answer a question of Yufei Zhao: vertex percolation at rate $p$ preserves the max-cut fraction whenever $p\\,e(G)/v(G)\\to\\infty$, and this condition is sharp. (iv) For bounded $r$-local quantum Hamiltonians of fixed local dimension, $O(\\varepsilon^{-3})$ uniformly sampled particles suffice to estimate the ground-state energy within $\\pm\\varepsilon n^r$, improving the previous $O(\\varepsilon^{-6}\\log(1/\\varepsilon))$ bound.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03468",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiaohui Bei"
    },
    {
      "name": "Wenhao Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/99061ym76ywkd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03468",
  "fulltext_url": null,
  "id": "99061ym76ywkd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03468",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03468",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/99061ym76ywkd.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:14Z",
      "native_id": "2611.03468",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03468"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Optimizer's Windfall: Bounding Hindsight Gain under Random Restrictions",
  "title_text": "The Optimizer's Windfall: Bounding Hindsight Gain under Random Restrictions",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:28Z",
  "url": "https://research.draconitic.com/paper/99061ym76ywkd",
  "venue": "arXiv",
  "versions": []
}