{
  "id": "1212.6027",
  "version": "v2",
  "published": "2012-12-25T11:38:51.000Z",
  "updated": "2014-09-05T13:09:37.000Z",
  "title": "Belief propagation for optimal edge cover in the random complete graph",
  "authors": [
    "Mustafa Khandwawala",
    "Rajesh Sundaresan"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/13-AAP981 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Applied Probability 2014, Vol. 24, No. 6, 2414-2454",
  "doi": "10.1214/13-AAP981",
  "categories": [
    "math.PR",
    "cs.DM",
    "cs.IT",
    "math.IT"
  ],
  "abstract": "We apply the objective method of Aldous to the problem of finding the minimum-cost edge cover of the complete graph with random independent and identically distributed edge costs. The limit, as the number of vertices goes to infinity, of the expected minimum cost for this problem is known via a combinatorial approach of Hessler and W\\\"{a}stlund. We provide a proof of this result using the machinery of the objective method and local weak convergence, which was used to prove the $\\zeta(2)$ limit of the random assignment problem. A proof via the objective method is useful because it provides us with more information on the nature of the edge's incident on a typical root in the minimum-cost edge cover. We further show that a belief propagation algorithm converges asymptotically to the optimal solution. This can be applied in a computational linguistics problem of semantic projection. The belief propagation algorithm yields a near optimal solution with lesser complexity than the known best algorithms designed for optimality in worst-case settings.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-12-25T11:38:51.000Z",
      "title": "Belief propagation for optimal edge-cover in the random complete graph",
      "abstract": "We apply the objective method of Aldous to the problem of finding the minimum cost edge-cover of the complete graph with random independent and identically distributed edge-costs. The limit, as the number of vertices goes to infinity, of the expected minimum cost for this problem is known via a combinatorial approach of Hessler and W\\\"astlund. We provide a proof of this result using the machinery of the objective method and local weak convergence, which was used to prove the \\zeta(2) limit of the random assignment problem. A proof via the objective method is useful because it provides us more information on the nature of the edges incident on a typical root in the minimum cost edge cover. We further show that a belief propagation algorithm converges asymptotically to the optimal solution. This finds application in a computational linguistics problem of semantic projection. The belief propagation algorithm yields a near optimal solution with lesser complexity than the known best algorithms designed for optimality in worst-case settings.",
      "comment": "arXiv admin note: text overlap with arXiv:math/0401388 by other authors",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-09-05T13:09:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05",
      "68Q87",
      "82B44"
    ],
    "keywords": [
      "random complete graph",
      "optimal edge-cover",
      "propagation algorithm converges",
      "objective method",
      "minimum cost edge cover"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.6027K"
    }
  }
}