{
  "abstract": "We prove a tight lower bound on symmetric LP formulations for the spanning tree polytope of the complete graph. More precisely, let $P_{ST}(K_n)$ be the spanning tree polytope of $K_n$. We show that, for all $n\\ge13$, every symmetric extended formulation for $P_{ST}(K_n)$ has at least $\\binom n3$ inequalities. Since the classical Martin formulation has a symmetric formulation of size $O(n^3)$, this gives \\[ \\operatorname{xcs}(P_{ST}(K_n))=Θ(n^3). \\] We also study outer relaxations of the spanning tree polytope of $K_n$ for maximization with nonnegative edge weights. We show that for every fixed approximation factor below $3/2$, symmetric LP formulations still require $Θ(n^3)$ inequalities for all sufficiently large $n$, whereas factor $3/2$ admits a formulation of size $O(n^2)$. Moreover, we provide an exact symmetric SDP formulation of size $O(n^2)$, measured by PSD matrix order, which implies $\\operatorname{xc}_{\\mathrm{psd}}(P_{ST}(K_n))= Θ(n^2)$. Thus symmetric LP and SDP sizes already differ for exact formulations, and for every fixed approximation factor $1\\leβ<3/2$ for all sufficiently large $n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.17017",
  "author_total": 1,
  "authors": [
    {
      "name": "Sebastian Pokutta"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/50xs6nxj5d9ba.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.17017",
  "fulltext_url": null,
  "id": "50xs6nxj5d9ba",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.17017",
  "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"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7164911416",
  "pdf_url": "https://arxiv.org/pdf/2606.17017",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/50xs6nxj5d9ba.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2606.17017",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.17017"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "(Approximate) Symmetric Extension Complexity of the Spanning Tree Polytope",
  "title_text": "(Approximate) Symmetric Extension Complexity of the Spanning Tree Polytope",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/50xs6nxj5d9ba",
  "venue": "arXiv",
  "versions": []
}