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SUMMARY:Dabeen Lee (이다빈)\, From coordinate subspaces over finite fields to ideal multipartite uniform clutters
DESCRIPTION:Take a prime power $q$\, an integer $n\geq 2$\, and a coordinate subspace $S\subseteq GF(q)^n$ over the Galois field $GF(q)$. One can associate with $S$ an $n$-partite $n$-uniform clutter $\mathcal{C}$\, where every part has size $q$ and there is a bijection between the vectors in $S$ and the members of $\mathcal{C}$. In this paper\, we determine when the clutter $\mathcal{C}$ is ideal\, a property developed in connection to Packing and Covering problems in the areas of Integer Programming and Combinatorial Optimization. Interestingly\, the characterization differs depending on whether $q$ is $2\,4$\, a higher power of $2$\, or otherwise. Each characterization uses crucially that idealness is a minor-closed property: first the list of excluded minors is identified\, and only then is the global structure determined. A key insight is that idealness of $\mathcal{C}$ depends solely on the underlying matroid of $S$. Our theorems also extend from idealness to the stronger max-flow min-cut property. As a consequence\, we prove the Replication and $\tau=2$ conjectures for this class of clutters. This is joint work with Ahmad Abdi (London School of Economics).
URL:https://dimag.ibs.re.kr/event/2023-08-29/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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