{
  "abstract": "We prove the all-exp property for the Traveling Salesman Problem with the $2$-opt neighborhood, and for $(3,2,2)$- and $(2,3,2)$-Maximum Constraint Assignment with the flip neighborhood. The all-exp property states that infinitely many instances have an initial solution from which every sequence of improving moves to a local optimum, and hence every run of local search, has exponential length. For the Traveling Salesman Problem, the all-exp property was previously known for every $k$-opt neighborhood with $k \\ge 3$, so our result for $2$-opt completes the picture for all $k$-kopt neighborhoods. In $(p,q,r)$-Maximum Constraint Assignment, each constraint involves at most $p$ variables, each variable occurs in at most $q$ constraints, and each domain has size $r$. Decreasing any of the parameters of $(3,2,2)$ or $(2,3,2)$ yields a problem in which every sequence of improving flips has polynomial length, so both Maximum Constraint Assignment results are tight. All three results settle long-standing open problems. Our proofs build on a common construction idea, which reveals a close connection between the local-search dynamics of the three problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31513",
  "author_total": 3,
  "authors": [
    {
      "name": "Sophia Heimann"
    },
    {
      "name": "Hung P. Hoang"
    },
    {
      "name": "Stefan Hougardy"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/btkpska3rn0jc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31513",
  "fulltext_url": null,
  "id": "btkpska3rn0jc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31513",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7223348823",
  "pdf_url": "https://arxiv.org/pdf/2610.31513",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/btkpska3rn0jc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.31513",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31513"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Closing All-Exp Gaps in Local Search for the Traveling Salesman Problem and Maximum Constraint Assignment",
  "title_text": "Closing All-Exp Gaps in Local Search for the Traveling Salesman Problem and Maximum Constraint Assignment",
  "type": "preprint",
  "updated_at": "2026-11-07T18:20:08Z",
  "url": "https://research.draconitic.com/paper/btkpska3rn0jc",
  "venue": "arXiv",
  "versions": []
}