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{
    "pk": 64911,
    "title": "Covering grids with multiplicity",
    "subtitle": null,
    "abstract": "Given a finite grid in \\(\\mathbb{R}^2\\), how many lines are needed to cover all but one point at least \\(k\\) times? Problems of this nature have been studied for decades, with a general lower bound having been established by Ball and Serra. We solve this problem for various types of grids, in particular showing the tightness of the Ball-Serra bound when one side is much larger than the other. In other cases, we prove new lower bounds that improve upon Ball-Serra and provide an asymptotic answer for almost all grids. For the standard grid \\(\\{0,\\ldots,n-1\\} \\times \\{0,\\ldots,n-1\\}\\), we prove nontrivial upper and lower bounds on the number of lines needed. To prove our results, we combine linear programming duality with some combinatorial arguments.\n \n \nMathematics Subject Classifications: 05B40, 52C15, 05D40\n \nKeywords: Grid covering, Alon-Füredi Theorem, combinatorial geometry, linear programming",
    "language": "en",
    "license": {
        "name": "Creative Commons Attribution 4.0",
        "short_name": "CC BY 4.0",
        "text": "Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.",
        "url": "https://creativecommons.org/licenses/by/4.0"
    },
    "keywords": [
        {
            "word": "Grid covering"
        },
        {
            "word": "Alon-Füredi Theorem"
        },
        {
            "word": "combinatorial geometry"
        },
        {
            "word": "linear programming"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/7p97h8mn",
    "frozenauthors": [
        {
            "first_name": "Anurag",
            "middle_name": "",
            "last_name": "Bishnoi",
            "name_suffix": "",
            "institution": "Delft Institute of Applied Mathematics, Technische Universiteit Delft, 2628 CD Delft, Netherlands",
            "department": ""
        },
        {
            "first_name": "Simona",
            "middle_name": "",
            "last_name": "Boyadzhiyska",
            "name_suffix": "",
            "institution": "School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, U.K.",
            "department": ""
        },
        {
            "first_name": "Shagnik",
            "middle_name": "",
            "last_name": "Das",
            "name_suffix": "",
            "institution": "Department of Mathematics, National Taiwan University, Taiwan",
            "department": ""
        },
        {
            "first_name": "Yvonne",
            "middle_name": "den",
            "last_name": "Bakker",
            "name_suffix": "",
            "institution": "Delft Institute of Applied Mathematics, Technische Universiteit Delft, 2628 CD Delft, Netherlands",
            "department": ""
        }
    ],
    "date_submitted": "2023-12-22T13:28:43Z",
    "date_accepted": "2023-12-22T13:28:43Z",
    "date_published": "2023-12-22T08:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64911/galley/49721/download/"
        }
    ]
}