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

Date and Time: Tuesday, November 23, 2010, 12:15 pm

Duration: This information is not available in the database

Location: OAT S15/S16/S17

Speaker: Marek Sulovský

Recent events on the conflict-free front.

A conflict-free coloring of a hypergraph H=(V,E) is a coloring of its vertices in such a way, that every nonempty hyperedge contains a vertex of unique color (within the heyperedge). This notion has been motivated by the frequency assignment problem (classical situation with mobile devices and antennae) and has been studied for several hypergraphs arising from geometric objects.

In this talk, we will look at the list variant of the problem. We show, that every hreditarily k-colorable hypergraph on n vertices can be conflict-free list-colored with list of size log_{1+1/k}(n)+1. Furthermore, it is possible to achieve, that the highest color (assuming some order of the colors) in every hyperedge is unique. This is already nontrivial for points and intervals in R. It for instance implies list-colorability of points w.r.t. intervals by log_2(n)+1 colors, points in R2 w.r.t. halfspaces by 1.71log_2(n)+1 colors and points in R2 w.r.t. disks by 2.41log_2(n)+1 colors.

Maybe a bit surprisingly, the result exactly (including the multiplicative constant) matches what is known for usual conflict-free coloring, although the proof for the list setting is more complicated.

Joint work with שכר סמורודינסקי and Πανοσ Χειλαρισ.

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