Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Tuesday, April 13, 2021, 12:15 pm

Duration: 30 minutes

Location: Zoom: conference room

Speaker: Bernd Gärtner

A New Combinatorial Property of Geometric Unique Sink Orientations

A unique sink orientation (USO) is an orientation of the n-dimensional hypercube graph, with the property that every subgraph induced by a nonempty face has a unique sink. In particular, there is a unique global sink, and a number of well-known optimization problems reduce to sink-finding in USOs. Such reductions have successfully been used in the past to gain new insights on a number of concrete optimization problems. Since the geometric USOs arising in this way are only a tiny fraction of all USOs, it is surprising how powerful the combinatorial abstraction still is.

If we had a better understanding of geometric USOs, we could further strengthen the abstraction, via additional combinatorial properties that certain classes of geometric USOs necessarily have. Some such properties have been found in the past, but nothing has happened over the last 10 years. In this talk, I present a new combinatorial property for a relevant class of geometric USOs. From every vertex of the USO, we derive a directed graph in a natural way, and the property is that all these graphs are acyclic.

This is joint work with Yuan Gao and Jourdain Lamperski (https://arxiv.org/abs/2008.08992)

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