PPAD-completeness of polyhedral versions of Sperner's Lemma

Abstract We show that certain polyhedral versions of Sperner’s Lemma, where the colouring is given explicitly as part of the input, are PPAD -complete. The proofs are based on two recent results on the complexity of computational problems in game theory: the PPAD -completeness of 2-player Nash, proved by Chen and Deng, and of Scarf’s Lemma, proved by Kintali. We give a strengthening of the latter result, show how colourings of polyhedra provide a link between the two, and discuss a special case related to vertex covers.

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