Survey: Colouring, constraint satisfaction, and complexity
Computer Science Review
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The problem of finding a fixed point of a nonexpansive mapping on a hypercube is that it has a polynomial time algorithm. In fact, it is known that one can find a 2-satisfiability characterization of the set of all fixed points in polynomial time. This implies that the problem of finding a vertex that is a common fixed point of several given nonexpansive mappings on a hypercube is that it has a polynomial time algorithm.We consider the problem of finding a vertex that is a common fixed point of several given nonexpansive mappings on a more general Cartesian product of graphs. For a single nonexpansive mapping, a known polynomial time algorithm finds a fixed point and a 2-satisfiability-like characterization of all fixed points. We introduce graphs with a farthest point property (also called apiculate graphs in [H. J. Bandelt and V. Chepoi, The Algebra of Metric Betweenness: Subdirect Representations, Retracts, and Axiomatics, manuscript]), and show that finding a common fixed point of several nonexpansive mappings on Cartesian products of such graphs involves using a polynomial time algorithm. We generalize this result to any family of graphs having a majority function.By contrast, the smallest graph (in the sense of having the fewest vertices, and the fewest edges of those having the fewest vertices) without the farthest point property is K2,3, and finding a vertex that is a fixed point of two given nonexpansive mappings (retractions) on a Cartesian product of graphs isomorphic to K2,3 is NP-complete. More generally, we exhibit an infinite family of graphs without the farthest point property giving NP-completeness. We show that for any family of graphs not having a majority function, the existence of a common fixed point of two nonexpansive mappings on Cartesian products of such graphs is NP-complete. This proves a dichotomy for the problem based on the existence of a majority function; a similar dichotomy is obtained for the special case of nonexpansive mappings that are retractions. Finally we characterize the families of chordal graphs corresponding to both dichotomies.