Algorithms for the fixed point property

Abstract

This survey exhibits various algorithms to decide the question if a given ordered set P has the fixed point property resp. if P has a fixed point free order-preserving self-map. While a depth-first search algorithm for a fixed point free map is easily written it is also quite inefficient. We discuss a reduction algorithm by Xia which can be used to speed up the search for a fixed point free self-map. The ideas used in creating this algorithm show close connections to two problems: the decision whether an ordered set has a fixed point free automorphism and the decision whether a given r-partite graph has an r-clique. The latter two problems are shown to be NP-complete using the work of Goddard, Lubiw and Williamson. The problem to decide whether a given finite ordered set has a fixed point free order-preserving self-map has recently been shown to be NP-complete, thus showing that the above close connection is not by accident. Retraction theorems leading to dismantling algorithms are another approach to the problem. We present the classical dismantling procedure by Rival and extensions by Fofanova, Li, Milner, Rutkowski and the author. These theorems give a polynomial algorithm to decide if an ordered set has the fixed point property for some nice classes of ordered sets (height 1, width 2), and structural insights for other classes (chain-complete ordered sets with no infinite antichains, sets of (interval) dimension 2). The related issue of uniqueness of cores gives an insight into Birkhoff's problem regarding cancellation of exponents. Walker's relational fixed point property for which the analogous problem has a very satisfying solution also is discussed. Another variation on the retraction theme is the use of algebraic topology in deriving fixed point theorems initiated by Baclawski and Björner and continued for example by Constantin and Fournier. After a primer on the basic concepts of (integer) homology we present their retraction/dismantling procedures, which always prove acyclicity of the associated simplicial complex and then the fixed point property via a Lefschetz-type fixed point theorem. Differences and similarities with the above combinatorial procedures are pointed out. The main problem in making these results accessible to entirely combinatorial proofs is the lack of a combinatorial/discrete analogue of the continuous concept of a weak (resp. deformation) retract. We also present a class of ordered sets without the fixed point property, such that all proper retracts have the fixed point property. This class is induced via triangulations of n-spheres. Finally, we include an indication how methods developed for work on the fixed point property can be used in other areas. For example, the arguments by Abian, Brown and Pelczar to prove that in a chain-complete ordered set existence of a point P with P ⩽ f( P) implies existence of a fixed point have recently found application to analysis in the work of Carl, Heikkilä, Lakhshmikantham and others.