Idealness of k-wise intersecting families
Abdi, A.
, Cornuéjols, G., Huynh, T. & Lee, D.
(2020).
Idealness of k-wise intersecting families.
In
Bienstock, D. & Zambelli, G.
(Eds.),
Integer Programming and Combinatorial Optimization - 21st International Conference, IPCO 2020, Proceedings
(pp. 1 - 12).
Springer Berlin / Heidelberg.
https://doi.org/10.1007/978-3-030-45771-6_1
, Cornuéjols, G., Huynh, T. & Lee, D.
(2020).
Idealness of k-wise intersecting families.
In
Bienstock, D. & Zambelli, G.
(Eds.),
Integer Programming and Combinatorial Optimization - 21st International Conference, IPCO 2020, Proceedings
(pp. 1 - 12).
Springer Berlin / Heidelberg.
https://doi.org/10.1007/978-3-030-45771-6_1
A clutter is k-wise intersecting if every k members have a common element, yet no element belongs to all members. We conjecture that every 4-wise intersecting clutter is non-ideal. As evidence for our conjecture, we prove it in the binary case. 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 As further evidence for our conjecture, we also note that it follows from an unpublished conjecture of Seymour from 1975.