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Methods of computing fixed points can be regarded as the culmination of many years of mathematical research, starting with Brouwer’s nonconstructive fixed point theorem in 1910. The breakthrough came in 1967 when Scarf succeeded in giving the first constructive proof of Brouwer’s fixed point theorem.
There are now a number of algorithms for computing fixed points using triangulation or primitive sets, based on Scarf’s concept, and complementary pivoting techniques. All these algorithms provide a constructive proof of Sperner’s Lemma for triangulation or a version of Sperner’s Lemma for primitive sets.
It is shown that they have a common combinatorial structure, which can be described, for instance, in terms of maximal chains with respect to a binary relation. This can be the basis for constructing new algorithms of similar type.