{
  "abstract": "A wiring diagram is an arrangement of $m$ pseudolines, every two of which cross exactly once. Reading its crossings from left to right encodes it as a reduced word of the permutation $w_0\\in S_m$ that reverses $1,\\dots,m$, unique up to commutations, and a braid move $σ_iσ_{i+1}σ_i\\leftrightarrowσ_{i+1}σ_iσ_{i+1}$ flips one of its triangles. We prove that deciding whether two wiring diagrams are within a given braid-move distance is NP-complete. Equivalently, deciding flip distance is NP-complete in the higher Bruhat order $B(m,2)$, for triangle flips of marked arrangements of pseudolines, and for rhombic tilings of a $2m$-gon; this answers an open question on flip graphs of pseudoline arrangements (SODA 2024). The reduction is from Vertex Cover: for every graph $G$ we construct two wiring diagrams at braid-move distance $|D|+2\\operatorname{VC}(G)$, where $\\operatorname{VC}(G)$ is the vertex cover number of $G$ and the Hamming distance $|D|$ is the number of triples of wires whose triangles are oriented differently. The construction glues copies of a seven-wire gadget whose braid-move distance exceeds its Hamming distance by $2$. Adding a line at infinity transfers the construction to rank-$3$ uniform oriented matroids, where two oriented matroids are adjacent in the mutation graph if their chirotopes differ in one basis; so deciding distance in the mutation graph is NP-complete in rank $3$. An iterate of Rambau's expansion of signotopes amplifies the excess and gives NP-hardness in every fixed rank $r\\ge4$, and by duality in every fixed corank at least $3$. The same instances show that flip distance in the higher Bruhat orders $B(n,k)$ is NP-complete for every fixed $k\\ge2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03808",
  "author_total": 1,
  "authors": [
    {
      "name": "Tilen Marc"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bkxnjx06zdcsq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03808",
  "fulltext_url": null,
  "id": "bkxnjx06zdcsq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03808",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7220681625",
  "pdf_url": "https://arxiv.org/pdf/2610.03808",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bkxnjx06zdcsq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.03808",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03808"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Computing distances in braid-move graphs, higher Bruhat orders, and oriented-matroid mutation graphs is NP-hard",
  "title_text": "Computing distances in braid-move graphs, higher Bruhat orders, and oriented-matroid mutation graphs is NP-hard",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/bkxnjx06zdcsq",
  "venue": "arXiv",
  "versions": []
}