A clutter is a finite family of finite sets where no set contains another one. Clutters form a very broad class of objects generalizing graphs and matroids in more than one way, and are also accompanied by notions of duality (more specifically, the blocking relation) and minor theory.
Most questions about clutters can be traced back to the study of three key parameters: the covering number, the fractional packing number, and the packing number, ordered from largest to smallest. The first two parameters are known to be equal, and efficiently computable, for the class of ideal clutters, an important class of clutters rooted in Combinatorial Optimization and Polyhedral Combinatorics.
In 1975, Seymour conjectured that for every ideal clutter, the fractional packing number can be attained not just by a fractional vector, but by a vector whose entries are dyadic rational numbers. As a first step, we prove this conjecture in the case when the fractional packing number is equal to 2; we also provide a quasi-polynomial time algorithm for finding the dyadic vector. Our proof techniques rely on the key insight that ideal clutters are so-called clean, i.e. they forbid two infinite classes of clutters as a minor, one coming from projective planes, and another coming from odd holes.
In this self-contained talk, after contextualizing Seymour’s conjecture, I shall motivate clean clutters and survey two important subclasses other than ideal clutters, namely, binary clutters (coming from binary matroids), and clutters without an intersecting family as a minor.
Based on joint work with Gérard Cornuéjols, Bertrand Guenin, and Levent Tunçel.
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