{
  "abstract": "We study tropical circuits with scalar multiplication gates, that is, algebraic circuits whose gates implement $\\max$, $+$, or multiplication with a positive constant. For such circuits, we prove exponential size lower bounds for computing maximum weight directed spanning trees and maximum weight bipartite perfect matchings. As a corollary, we obtain an exponential size separation between monotone and non-monotone maxout neural networks, which generalize the popularly used ReLU neural networks. One conclusion from this is that neural network models with enforced convexity constraints, such as input-convex neural networks (ICNNs), sometimes need to be exponentially larger than their unrestricted counterparts in order to express the same functions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.11540",
  "author_total": 2,
  "authors": [
    {
      "name": "Christoph Hertrich"
    },
    {
      "name": "Moritz Stargalla"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cd54sg2b5w7c3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.11540",
  "fulltext_url": null,
  "id": "cd54sg2b5w7c3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.11540",
  "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.CC"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
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  "openalex": "W7168279256",
  "pdf_url": "https://arxiv.org/pdf/2607.11540",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cd54sg2b5w7c3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2607.11540",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.11540"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Tropical Circuits with Scalar Multiplication Gates",
  "title_text": "Tropical Circuits with Scalar Multiplication Gates",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/cd54sg2b5w7c3",
  "venue": "arXiv",
  "versions": []
}