A Characterization of the Realizable Matou\v{s}ek Unique Sink Orientations

The Matou\v{s}ek LP-type problems were used by Matou\v{s}ek to show that the Sharir-Welzl algorithm may require at least subexponential time. Later, G\"artner translated this result into the language of Unique Sink Orientations (USOs) and introduced the Matou\v{s}ek USOs, the USOs equivalent to Matou\v{s}ek's LP-type problems. He further showed that the Random Facet algorithm only requires quadratic time on the realizable subset of the Matou\v{s}ek USOs, but without characterizing this subset. In this paper, we deliver this missing characterization and also provide concrete realizations for all realizable Matou\v{s}ek USOs. Furthermore, we show that the realizable Matou\v{s}ek USOs are exactly the orientations arising from simple extensions of cyclic-P-matroids.