{
  "abstract": "For many optimization problems on random graphs, most prominently the largest independent set problem, natural algorithms reach only half the optimum, and such factor-two gaps are known to be sharp within broad algorithmic classes. We prove such a gap for finding large induced matchings in Erdős-Rényi random graphs. In the worst case, requiring a matching to be induced turns a polynomial-time solvable problem into an NP-hard one; our results reveal an average-case counterpart. In $G(n,p)$ with $C/n\\le p\\le 1-κ$, the largest induced matching has size $(1+o(1))L$ with high probability, where $L=\\log_{1/(1-p)}(np)$. A simple online greedy algorithm finds an induced matching of size $(1/2-ε)L$, while no online algorithm, even with adaptive inspection order and unbounded computation, finds one of size $(1/2+ε)L$ except with probability $\\exp(-Ω(L\\log(np)))$. In the sparse regime $p=d/n$, low-degree polynomial algorithms exhibit the same sharp threshold. Our guarantees rest on a maximality principle: with high probability, every maximal induced matching is half-optimal. Greedy outputs a maximal induced matching and our low-degree algorithm a near-maximal one. We bypass the delicate coupling arguments classically used to analyze greedy since Karp. For low-degree hardness, we adapt Wein's overlap gap argument, with forbidden structures that track overlaps between endpoint sets rather than edge sets, since a new edge may join two previously used endpoints and thus encounter no fresh randomness. For online hardness, we give a new and significantly simpler argument that avoids the correlated instances of prior OGP-based proofs: at a suitable stopping time, all pairs involving uninspected vertices remain unrevealed, so a direct first-moment bound suffices.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40369",
  "author_total": 5,
  "authors": [
    {
      "name": "Abhishek Dhawan"
    },
    {
      "name": "Eren C. Kızıldağ"
    },
    {
      "name": "Neeladri Maitra"
    },
    {
      "name": "Cheng Mao"
    },
    {
      "name": "Timothy L. H. Wee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dse65htpcrzb3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40369",
  "fulltext_url": null,
  "id": "dse65htpcrzb3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40369",
  "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.DM"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40369",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dse65htpcrzb3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.40369",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40369"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Half-Optimal Induced Matchings in Random Graphs: Online and Low-Degree Algorithms",
  "title_text": "Half-Optimal Induced Matchings in Random Graphs: Online and Low-Degree Algorithms",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:51Z",
  "url": "https://research.draconitic.com/paper/dse65htpcrzb3",
  "venue": "arXiv",
  "versions": []
}