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Ideal clutters that do not pack. (English) Zbl 1435.90117
Summary: For a clutter \(\mathcal{C}\) over ground set \(E\), a pair of distinct elements \(e, f \in E\) are coexclusive if every minimal cover contains at most one of them. An identification of \(\mathcal{C} \) is another clutter obtained after identifying coexclusive elements of \(\mathcal{C} \). If a clutter is nonpacking, then so is any identification of it. Inspired by this observation, and impelled by the lack of a qualitative characterization for ideal minimally nonpacking (mnp) clutters, we reduce ideal mnp clutters even further by taking their identifications. In doing so, we reveal chains of ideal mnp clutters, demonstrate the centrality of mnp clutters with covering number two, as well as provide a qualitative characterization of irreducible ideal mnp clutters with covering number two. At the core of this characterization lies a class of objects, called marginal cuboids, that naturally give rise to ideal nonpacking clutters with covering number two. We present an explicit class of marginal cuboids, and show that the corresponding clutters have one of \(Q_6, Q_{2, 1}, Q_{10}\) as a minor, where \(Q_6, Q_{2, 1}\) are known ideal mnp clutters, and \(Q_{10}\) is a new ideal mnp clutter.

90C27 Combinatorial optimization
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