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The Matou\v{s}ek LP-type problems were used by Matou\v{s}ek to show that theSharir-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 toMatou\v{s}ek's LP-type problems. He further showed that the Random Facetalgorithm only requires quadratic time on the realizable subset of theMatou\v{s}ek USOs, but without characterizing this subset. In this paper, wedeliver this missing characterization and also provide concrete realizationsfor all realizable Matou\v{s}ek USOs. Furthermore, we show that the realizableMatou\v{s}ek USOs are exactly the orientations arising from simple extensionsof cyclic-P-matroids.

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