{
  "abstract": "The Ungar game, introduced by Defant et al., is a combinatorial game played on a finite lattice, which originates from a geometric transformation introduced by Ungar to resolve Scott's slope problem. When played on a distributive lattice $L$, the Ungar game reduces to repeatedly choosing and removing a non-empty set of maximal elements from a finite poset $P$, where $L$ is represented as the lattice $J(P)$ of the order ideals of $P$ by Birkhoff's representation theorem. Defant et al. raised the open question of whether determining the outcome of the Ungar game on $J(P)$ is PSPACE-complete with respect to $|P|$. In this paper, we settle this open question in the affirmative. More precisely, we prove that the problem is PSPACE-complete even when restricted to posets $P$ with maximum chain length $l(P) \\le 3$ such that the underlying graph of the Hasse diagram of $P$ is bipartite, where the length of a chain is defined as the number of elements in the chain minus one. Furthermore, we show that the problem is NP-complete for $l(P) \\le 2$, whereas the case $l(P) \\le 1$ is known to belong to LOGSPACE.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11017",
  "author_total": 1,
  "authors": [
    {
      "name": "Kengo Hashimoto"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e8ryqmhhnqq9q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11017",
  "fulltext_url": null,
  "id": "e8ryqmhhnqq9q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11017",
  "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",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7221197567",
  "pdf_url": "https://arxiv.org/pdf/2610.11017",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e8ryqmhhnqq9q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.11017",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11017"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Computational Complexity of the Ungar Games on Distributive Lattices",
  "title_text": "The Computational Complexity of the Ungar Games on Distributive Lattices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/e8ryqmhhnqq9q",
  "venue": "arXiv",
  "versions": []
}