A random bit-flipping method for seeking agreement

Recently Papadimitriou has proposed a randomised ‘bit flipping’ method for solving the 2-satisfiability problem; and the author has proposed a randomised recolouring method which, given a 3-colourable graph, finds a 2-colouring of the vertices so that no triangle is monochromatic. Both methods involve finding a ‘bad’ configuration (unsatisfied clause, monochromatic triangle) and randomly changing one of the bits involved. In this paper we see how these problems and methods fit naturally in a more general geometrical context in which we seek a vector which ‘agrees’ with a given collection of vectors; and we propose a simple ‘bit-flipping’ method for the more general problem, which extends the solution methods for the two problems mentioned above. Further, we consider deterministic methods to handle such problems, and in particular we see how to solve the above ‘triangle problem’ for 3colourable graphs deterministically in polynomial time.