Abstract
A d-interval is a union of at most d disjoint closed intervals on a fixed line. Tardos [14] and the second author [11] used topological tools to bound the transversal number τ of a family H of d-intervals in terms of d and the matching number ν of H. We investigate the weighted and fractional versions of this problem and prove upper bounds that are tight up to constant factors. We apply both a topological method and an approach of Alon [1]. For the use of the latter, we prove a weighted version of Turán’s theorem. We also provide proofs of the upper bounds of [11] that are more direct than the original proofs.
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Aharoni, R., Kaiser, T. & Zerbib, S. Fractional covers and matchings in families of weighted d-intervals. Combinatorica 37, 555–572 (2017). https://doi.org/10.1007/s00493-016-3174-7
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DOI: https://doi.org/10.1007/s00493-016-3174-7