Abstract
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 kge 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 As further evidence for our conjecture, we also note that it follows from an unpublished conjecture of Seymour from 1975.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.
Highlights
Let V be a finite set of elements, and let C be a family of subsets of V called members
By the theory of totally dual integral linear systems, for every max-flow min-cut (MFMC) clutter, the primal (P) admits an integral optimal solution for every cost vector w ∈ Z+V [16]. This is why the class of MFMC clutters is a natural host to many beautiful min–max theorems in Combinatorial Optimization [12]
[31] We propose a line of attack for tackling Conjecture 3 via a deep connection to projective geometries over the two-element field, objects that give rise to k-wise intersecting clutters We discuss two applications of Theorem 4, one to uniform cycle covers in graphs, another to quarter-integral packings of value two in ideal binary clutters
Summary
Let V be a finite set of elements, and let C be a family of subsets of V called members. By the theory of totally dual integral linear systems, for every MFMC clutter, the primal (P) admits an integral optimal solution for every cost vector w ∈ Z+V [16]. Q6 := {{1, 3, 6}, {1, 4, 5}, {2, 3, 5}, {2, 4, 6}}, whose elements are the edges and whose members are the triangles of K4, is an intersecting clutter that is ideal [28]. Q6 is the smallest intersecting clutter which is ideal [1, Proposition 1.2]. Conjecture 3 There exists an integer k ≥ 4 such that every k-wise intersecting clutter is non-ideal. Theorem 4 Every 4-wise intersecting binary clutter is non-ideal. Proposition 5 There exists an ideal 3-wise intersecting binary clutter.
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