Fractional covers and matchings in families of weighted d-intervals

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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References

  1. [1]

    N. Alon: Piercing d-intervals, Disc. Comput. Geom. 19 (1998), 333–334.

    MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    N. Alon: Covering a hypergraph of subgraphs, Discrete Mathematics 257 (2002), 249–254.

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    E. Berger: KKM — A topological approach for trees, Combinatorica 25 (2004), 1–18.

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    R. L. Brooks: On coloring the nodes of a network, Proc. Cambridge Phil. Soc. 37 (1941), 194–197.

    Article  MATH  Google Scholar 

  5. [5]

    T. Gallai: Graphen mit triangulierbaren ungeraden Vielecken, Magyar Tud. Ak. Mat. Kut. Int. Közl. 7 (1962), 3–36 (in German).

    MathSciNet  MATH  Google Scholar 

  6. [6]

    Z. Füredi: Maximum degree and fractional matchings in uniform hypergraphs, Com-binatorica 1 (1981), 155–162.

    MathSciNet  Google Scholar 

  7. [7]

    A. Gyárfás: On the chromatic number of multiple interval graphs and overlap graphs, Discrete Mathematics 55 (1985), 161–166.

    MathSciNet  Article  MATH  Google Scholar 

  8. [8]

    A. Gyárfás: Combinatorics of intervals, Lecture notes, 2009.

    Google Scholar 

  9. [9]

    A. Gyárfás and J. Lehel: A Helly-type problem in trees, in: P. Erdős, A. Rényi and V.T. Sós, eds., Combinatorial Theory and its Applications, North-Holland, Amsterdam, 571–584, 1970.

    Google Scholar 

  10. [10]

    A. Gyárfás and J. Lehel: Covering and coloring problems for relatives of intervals, Discrete Mathematics 55 (1985), 167–180.

    MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    T. Kaiser: Transversals of d-intervals, Disc. Comput. Geom. 18 (1997), 195–203.

    MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    H. Komiya: A simple proof of K-K-M-S theorem, Econ. Theory 4 (1994), 463–466.

    Article  MATH  Google Scholar 

  13. [13]

    J. Matoušek: Lower bounds on the transversal numbers of d-intervals, Disc. Comput. Geom. 26 (2001), 283–287.

    MathSciNet  Article  MATH  Google Scholar 

  14. [14]

    G. Tardos: Transversals of 2-intervals, a topological approach, Combinatorica 15 (1995), 123–134.

    MathSciNet  Article  MATH  Google Scholar 

  15. [15]

    G. Tardos: Transversals of d-intervals — comparing three approaches, in: European Congress of Mathematics, Vol. II, Budapest, 1996, Progress in Mathematics 169, Birkhäuser, Basel, 234–243, 1998.

    Google Scholar 

  16. [16]

    L. S. Shapley: On balanced games without side payments, in: T. C. Hu and S. M. Robinson (eds.), Mathematical Programming, Math. Res. Center Publ. 30, Academic Press, New York, 261–290, 1973.

    Google Scholar 

  17. [17]

    R. G. Stanton, D. D. Cowan and L. O. James: Some results on path numbers, Proc. Louisiana Conf. on Combinatorics, Graph Theory and Computing, 112–135, 1970.

    Google Scholar 

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Correspondence to Ron Aharoni.

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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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Mathematics Subject Classification (2000)

  • 05D15
  • 05C72