{
  "abstract": "Given a bipartite graph $G$, we denote by $g(G)$ the largest product $|A||B|$ over pairs of vertex sets $A$ and $B$, one on each side, with no edge between them. These pairs are the bicliques of the bipartite complement of $G$; hence, computing $g(G)$ is the maximum-edge biclique problem, which is NP-hard, as shown by Peeters in 2003. Gillis and Glineur characterized it by a continuous quadratic program, in which a large enough penalty on the edges of $G$ makes the global maximizers correspond to the maximum pairs and the local maximizers to the maximal ones. We propose a new threshold for the penalty, which depends only on $g(G)$. For any penalty at or above it, the global maximizers correspond exactly to the maximum pairs, and every local maximizer with positive value corresponds to a maximal pair. This threshold is never larger than theirs and is sharp up to an additive $1/2$. Spectral and semidefinite bounds give computable penalties; in experiments, a simple local ascent succeeds far more often with the semidefinite one than with the earlier thresholds, although on our graphs that bound already reveals $g(G)$. The penalty formulation, whose optimal value is $\\sqrt{g(G)}/2$, is also a copositive formulation of $g(G)$. Strengthening a semidefinite bound of Laurent, Polak and Vargas with the completely positive cone gives exactly $\\sqrt{g(G)}/2$, while the doubly nonnegative strengthening is already inexact on six vertices. Finally, Pólya's linear programming approximations of the copositive cone become finite at an explicit level and converge to $\\sqrt{g(G)}/2$ without ever being exact.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07239",
  "author_total": 2,
  "authors": [
    {
      "name": "Youssouf Emine"
    },
    {
      "name": "Bissan Ghaddar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ekx6fkg32tz2s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07239",
  "fulltext_url": null,
  "id": "ekx6fkg32tz2s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07239",
  "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/2611.07239",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ekx6fkg32tz2s.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07239",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07239"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Exact penalties for maximum-edge bicliques in bipartite graphs",
  "title_text": "Exact penalties for maximum-edge bicliques in bipartite graphs",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:47Z",
  "url": "https://research.draconitic.com/paper/ekx6fkg32tz2s",
  "venue": "arXiv",
  "versions": []
}