{
  "abstract": "For a graph G, the Boolean quadric polytope P(G) is the convex hull of the binary solutions of $y_{ij}=x_ix_j$ for $ij\\in E(G)$, and Q(G) is its standard linear relaxation. For series-parallel graphs, Q(G) together with the odd-cycle inequalities describes P(G). Lee and Skipper showed vol(Q(G)) is polynomial-time computable for bounded treewidth and gave a closed formula for vol(P(G)) when G is a cycle. We resolve their question of giving an efficient algorithm for vol(P(G)) on series-parallel graphs. With $d=|V(G)|+|E(G)|$, we compute it with $O(d^7)$ arithmetic operations, and $O(d^5)$ if G is a cactus. The algorithm is a dynamic program over the series-parallel decomposition. The same framework computes vol(Q(G)) (hence the number of linear extensions of the incidence poset of G) with $O(d^4)$ operations, and $O(d^3)$ for cacti. With every $x_v$ fixed at 1/2, the recursion reduces to convolutions of univariate polynomials and computes the cut polytope volume of every series-parallel graph with m edges in $O(m^3)$ operations. We also study how much of Q(G) the polytope P(G) occupies. Short odd cycles matter more than long ones: in every graph, the odd-cycle inequalities of a cycle of length $\\ell$ cut off at most a fraction $2^{\\ell-1}/\\ell!$ of Q(G). Hence vol(P(G))/vol(Q(G)) $\\ge 1-\\sum_C 2^{|C|-1}/|C|!$ for series-parallel G, with C ranging over its cycles. The ratio does not factor over cycles sharing a vertex: in a flower of $k$ copies of $C_\\ell$ it decays like $ρ_\\ell^k$ for an explicit rational $ρ_\\ell$ below the ratio of $C_\\ell$. Nevertheless, the small fractions cut off by many long cycles compound: in the worst case, triangle inequalities, or odd-cycle inequalities up to any fixed length, close no fixed fraction of the gap between Q(G) and P(G), and the volume ratio can be exponentially small in d.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00734",
  "author_total": 1,
  "authors": [
    {
      "name": "Jon Lee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7g4z72naqz1x7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00734",
  "fulltext_url": null,
  "id": "7g4z72naqz1x7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00734",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.00734",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7g4z72naqz1x7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.00734",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00734"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Exact volume computation for Boolean quadric polytopes of series-parallel graphs",
  "title_text": "Exact volume computation for Boolean quadric polytopes of series-parallel graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/7g4z72naqz1x7",
  "venue": "arXiv",
  "versions": []
}