Cuboids, a class of clutters

Abdi, Ahmad ORCID: 0000-0002-3008-4167, Cornuéjols, Gérard, Guričanová, Natália and Lee, Dabeen (2020) Cuboids, a class of clutters. Journal of Combinatorial Theory, Series B, 142. 144 - 209. ISSN 0095-8956

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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.

Item Type:	Article
Official URL:	https://www.journals.elsevier.com/journal-of-combi...
Additional Information:	© 2019 Elsevier Inc
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	06 Nov 2019 13:06
Last Modified:	20 Dec 2024 00:37
URI:	http://eprints.lse.ac.uk/id/eprint/102395

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