{
  "abstract": "We study nonpreemptive scheduling on a single machine with release dates, due dates, positive job weights, and a common processing time. The objective is to minimize total weighted tardiness. Although closely related equal-processing-time problems admit polynomial-time algorithms, the complexity of this problem has remained open in the literature since 2010. We prove that its decision version is strongly NP-complete, even when every job can meet its due date if processed immediately upon release. The reduction is from unweighted MAX-CUT and uses a quadratic number of jobs with polynomially bounded numerical data. Its main ingredient is a constructive normalization theorem that converts every sufficiently inexpensive feasible schedule into a binary choice for each graph vertex; after normalization, total weighted tardiness equals a constant minus a scaled cut value. We also give a deterministic polynomial-time phase-grid assignment algorithm for the shifted objective $Φ=F+p\\sum_jw_j$, where $F$ is total weighted tardiness. The algorithm enumerates at most $N$ release-date residues modulo $p$, solves one minimum-cost assignment problem for each residue, and returns the best phase-grid schedule. It runs in $O(N^5)$ arithmetic operations and achieves the tight ratio $3/2-1/(2N)$ for this algorithm. Because the added term $p\\sum_jw_j$ is independent of how the jobs are scheduled, the shifted and original objectives have exactly the same optimal schedules. However, the approximation guarantee applies to the shifted objective; for the original objective, the analysis provides an additive bound. Thus, the paper both resolves the long-standing complexity question and provides a complementary worst-case guarantee for the phase-grid assignment algorithm.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28751",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhi-Long Chen"
    },
    {
      "name": "Nicholas G. Hall"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5vfb25gz8s5dd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28751",
  "fulltext_url": null,
  "id": "5vfb25gz8s5dd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28751",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CC"
    ],
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2209|Scheduling and Optimization Algorithms"
    ]
  ],
  "openalex": "W7214368679",
  "pdf_url": "https://arxiv.org/pdf/2609.28751",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5vfb25gz8s5dd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28751",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28751"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Strong NP-Hardness and Approximation Algorithm for Weighted Tardiness with Release Dates and Identical Processing Times",
  "title_text": "Strong NP-Hardness and Approximation Algorithm for Weighted Tardiness with Release Dates and Identical Processing Times",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/5vfb25gz8s5dd",
  "venue": "arXiv",
  "versions": []
}