Cuboids, a class of clutters

Abstract

The τ = 2 Conjecture, the Replication Conjecture and the f-Flowing Conjecture, and the classification of binary matroids with the sums of circuits property are foundational to Clutter Theory and have far-reaching consequences in Combinatorial Optimization, Matroid Theory and Graph Theory. We prove that these conjectures and result can equivalently be formulated in terms of cuboids, which form a special class of clutters. Cuboids are used as means to (a) manifest the geometry behind primal integrality and dual integrality of set covering linear programs, and (b) reveal a geometric rift between these two properties, in turn explaining why primal integrality does not imply dual integrality for set covering linear programs. Along the way, we see that the geometry supports the τ = 2 Conjecture. Studying the geometry also leads to over 700 new ideal minimally non-packing clutters over at most 14 elements, a surprising revelation as there was once thought to be only one such clutter. CONTENTS AHMAD ABDI, GÉRARD CORNUÉJOLS, NATÁLIA GURICANOVÁ, AND DABEEN LEE 5.1. Products and coproducts of clutters 31 5.2. Products and coproducts of sets 32 5.3. Reflective products of sets 33 5.4. Strict connectivity and the R k,1 's 36 6. The spectrum of strictly non-polar sets of constant degree 37 6.1. Proof of Theorem 1.20 37 6.2. Generating strictly non-polar sets of degree at most 4 43 7. Concluding remarks and open questions 46 Acknowledgements 48 THE STRICTLY NON-POLAR SETS OF DEGREE AT MOST 4 AND DIMENSION AT MOST 7 The strictly non-polar sets of degree at most 4 and dimension at most 7, ordered according to (degree, dimension) and by size (0,3) ("100" "010" "001" "111")

Idealness of k-wise Intersecting Families

“…In particular, every member of cuboid(S) has size n (hence cuboid(S) is a clutter) and τ (cuboid(S)) ≤ 2. Cuboids were introduced in [5] and further studied in [1].…”

“…Let conv(S) denote the convex hull of S. An inequality of the form ∑ i∈I x i + ∑ j∈J (1 − x j ) ≥ 1, for some disjoint I, J ⊆ [n], is called a generalized set covering inequality [8]. The set S is cube-ideal if every facet of conv(S) is defined by x i ≥ 0, x i ≤ 1, or a generalized set covering inequality [1].…”

Ideal Clutters That Do Not Pack

“…We may assume that (p, q, r) = (1,2,3). Consider now the subcube of implying in turn that C has one of Q 3 , Q 6 as a cuboid minor.…”

“…showed that these clutters possess a lot of structure, and his structure is qualitative in the sense that it explains why such clutters are non-ideal and do not pack. 1 Therefore, in light of the result above, we consider non-ideal mnp clutters well-understood, and focus on ideal mnp clutters.…”

Delta Minors, Delta Free Clutters, and Entanglement

“…A clutter C is binary if, for all members C 1 , C 2 , C 3 , the symmetric difference C 1 △C 2 △C 3 contains a member. 2 It can be readily checked that if a clutter is binary, then so is every minor of it [18,22]. It is known that a 1 A − B := {a ∈ A : a / ∈ B} 2 A 1 △ • • • △A k is the set of elements that belong to an odd number of the A i 's.…”