Idealness of k-wise intersecting families

Ahmad Abdi, Gérard Cornuéjols, Tony Huynh, Dabeen Lee

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5 Citations (Scopus)


A clutter is k-wise intersecting if every k members have a common element, yet no element belongs to all members. We conjecture that, for some integer k≥ 4 , every k-wise intersecting clutter is non-ideal. As evidence for our conjecture, we prove it for k= 4 for the class of binary clutters. Two key ingredients for our proof are Jaeger’s 8-flow theorem for graphs, and Seymour’s characterization of the binary matroids with the sums of circuits property. As further evidence for our conjecture, we also note that it follows from an unpublished conjecture of Seymour from 1975. We also discuss connections to the chromatic number of a clutter, projective geometries over the two-element field, uniform cycle covers in graphs, and quarter-integral packings of value two in ideal clutters.

Original languageEnglish
Pages (from-to)29-50
Number of pages22
JournalMathematical Programming
Issue number1-2
Publication statusPublished - Mar 2022


  • 8-Flow theorem
  • Binary clutters
  • Ideal clutters
  • k-wise intersecting families
  • Projective geometries
  • Quarter-integral packings
  • Sums of circuits property
  • Idealness of k-wise Intersecting Families

    Abdi, A., Cornuéjols, G., Huynh, T. & Lee, D., 1 Jan 2020, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics): 21st International Conference on Integer Programming and Combinatorial Optimization, IPCO 2020; London; United Kingdom; 8 June 2020 through 10 June 2020. Bienstock, D. & Zambelli, G. (eds.). Cham Switzerland: Springer-Praxis, Vol. 12125. p. 1-12 12 p. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); vol. 12125 LNCS).

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    5 Citations (Scopus)

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