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Series B

Idealness of k-wise intersecting families Ahmad Abdi1

· Gérard Cornuéjols2 · Tony Huynh3 · Dabeen Lee4

Received: 23 June 2020 / Accepted: 27 October 2020 © The Author(s) 2020

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 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 As further evidence for our conjecture, we also note that it follows from an unpublished conjecture of Seymour from 1975.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. Keywords Ideal clutters · k-wise intersecting families · Binary clutters · Quarter-integral packings · Sums of circuits property · 8-Flow theorem · Projective geometries

An extended abstract of this work was published in the proceedings of the 21st conference in Integer Programming and Combinatorial Optimization [2]. This work was supported by NSERC PDF grant 516584-2018, ONR grant 00014-18-12129, the Australian Research Council, ERC grant FOREFRONT (Grant Agreement No. 615640), and the Institute for Basic Science (IBS-R029-C1).

B

Ahmad Abdi [email protected] Gérard Cornuéjols [email protected] Tony Huynh [email protected] Dabeen Lee [email protected]

1

Department of Mathematics, London School of Economics and Political Science, London, UK

2

Tepper School of Business, Carnegie Mellon University, Pittsburgh, USA

3

School of Mathematics, Monash University, Melbourne, Australia

4

Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea

123

A. Abdi et al.

Mathematics Subject Classification 90C57 · 05B40

1 Introduction Let V be a finite set of elements, and let C be a family of subsets of V called members. The family C is a clutter over ground set V , if no member contains another one [15]. A cover of C is a subset B ⊆ V such that B ∩ C = ∅ for all C ∈ C. Consider for V the dual pair of linear programs w ∈ Z+ min  w x (P) s.t. (xu : u ∈ C) ≥ 1 ∀C ∈ C x ≥0 max  1 y (D) s.t. (yC : u ∈ C ∈ C) ≤ wu ∀u ∈ V y ≥ 0. If the dual (D) has an integral optimal solution for every right-hand-side vector w ∈ V , then C is said to have the max-flow min-cut (MFMC) property [10]. By the theory Z+ of totally dual integral linear systems, for every MFMC clutter, the primal (P) also V [16]. This is why admits an integral optimal solution for every cost vector w ∈ Z+ the class of MFMC clutters is a natural host to many beautiful min–max theorems in Combinatorial Optimization [12]. Let us elaborate. The packing number of C, denoted ν(C), is the maximum number of pairwise disjoint members. Note that ν(C) is equal to the m