Identically Self-blocking Clutters

Abdi, Ahmad; Cornuéjols, Gérard; Lee, Dabeen

doi:10.1007/978-3-030-17953-3_1

Cited by 3 publications

(4 citation statements)

References 21 publications

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“…A result that must be noted in the context of Conjecture 9 is the following: For every ideal clutter C where every member and also every cover has cardinality at least two, either χ(C) ≤ 2 or χ(b(C)) ≤ 2 (or both). This result was proved recently, and the proof relied on Gauge Duality in Quadratic Programming [3].…”

Section: Proposition 11mentioning

confidence: 85%

“…Similarly, we may assume e / ∈ C i 3 for i ∈ [2]. But now {C 1 j C 2 j : j ∈ [3]} is a 3-cycle cover of M. For (3), suppose E 1 ∩ E 2 = {e, f , g}. Since {e, f , g} is a cocircuit of both M 1 , M 2 , and since cocircuits and circuits of a binary matroid have an even number of elements in common, |C i j ∩ {e, f , g}| ∈ {0, 2} for all i, j.…”

Section: Claim 1 Both F 7 and M(v 8 ) Have 3-cycle Coversmentioning

confidence: 99%

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Idealness of k-wise intersecting families

et al. 2020

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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 $$k\ge 4$$ k ≥ 4 , every k-wise intersecting clutter is non-ideal. As evidence for our conjecture, we prove it for $$k=4$$ 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 As further evidence for our conjecture, we also note that it follows from an unpublished conjecture of Seymour from 1975.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.

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Section: Proposition 11mentioning

confidence: 85%

Section: Claim 1 Both F 7 and M(v 8 ) Have 3-cycle Coversmentioning

confidence: 99%

Idealness of k-wise intersecting families

et al. 2020

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“…that describes yet another mechanism of majority voting, 1 but this decomposition has no essential meaning from the linear algebraic viewpoint, since 1 Let 2 [t] denote the power set (i.e., the family of all subsets) of Et. Recall that maps…”

Section: Introductionmentioning

confidence: 99%

Pattern Recognition on Oriented Matroids: Symmetric Cycles in the Hypercube Graphs. V

Matveev

2021

Preprint

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We consider decompositions of topes of the oriented matroid realizable as the arrangement of coordinate hyperplanes in R 2 t , with respect to a distinguished symmetric 2 • 2 t -cycle in its hypercube graph of topes H(2 t , 2). We seek interpretations of such decompositions in the context of subset families on the ground set Et := {1, . . . , t} and of the families of their blocking sets, in the context of clutters on Et and of their blockers. Contents 1. Introduction Decomposing 2. Topes, their relabeled opposites, and decompositions Blocking 3. Increasing families of blocking sets, and blockers: Set covering problems 4. Families of subsets of the ground set E t : Characteristic vectors and characteristic topes 5. Increasing families of blocking sets, and blockers: Characteristic vectors and characteristic topes 5.1. A clutter {{a}} 5.1.1. The principal increasing family of blocking sets B({{a}}) = {{a}} 5.1.2. The blocker B({{a}}) = {{a}} 5.1.3. More on the principal increasing family B({{a}}) = {{a}} 5.2. A clutter {A} 5.2.1. The increasing family of blocking sets B({A}) = {{a} : a ∈ A} 5.2.2. The blocker B({A}) = {{a} : a ∈ A} 5.2.3. More on the increasing families {A} and B({A}) 5.3. A clutter A := {A 1 , . . . , A α } 5.3.1. The increasing family of blocking sets B(A) 5.3.2. The blocker B(A) 5.3.3. More on the increasing families A and B(A)

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“…In fact, Q 6 is the smallest intersecting clutter which is not ideal ([1], Proposition 1.2). It is worth pointing out that if a clutter and its blocker are both intersecting, then the clutter must be nonideal [3].…”

Section: Introductionmentioning

confidence: 99%

Idealness of k-wise Intersecting Families

Abdi

Cornuéjols

Huynh

et al. 2020

Integer Programming and Combinatorial Optimization

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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. arXiv:1912

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Identically Self-blocking Clutters

Cited by 3 publications

References 21 publications

Idealness of k-wise intersecting families

Idealness of k-wise intersecting families

Pattern Recognition on Oriented Matroids: Symmetric Cycles in the Hypercube Graphs. V

Idealness of k-wise Intersecting Families

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