Flexible constraint satisfiability and a problem in semigroup theory

We examine some flexible notions of constraint satisfaction, observing some relationships between model theoretic notions of universal Horn class membership and robust satisfiability. We show the NP-completeness of 2-robust monotone 1-in-3 3SAT in order to give very small examples of finite algebras with NP-hard variety membership problem. In particular we give a 3-element algebra with this property, and solve a widely stated problem by showing that the 6-element Brandt monoid has NP-hard variety membership problem. These are the smallest possible sizes for a general algebra and a semigroup to exhibit NP-hardness for the membership problem of finite algebras in finitely generated varieties. In a number of computational situations, the task is not to find if there is a single solution to an instance, but rather to find whether there is a sufficiently broad family of solutions to witness various separation conditions on variables. The following example is illustrative of idea. Recall that a 3-colouring of a graphG is a homomorphism fromG to the complete graphK3. Equivalently, there is a map to the set {0, 1, 2} such that adjacent vertices have different colours. Graph colouring places no restriction on what is to happen to non-adjacent vertices. What happens then if we ask that a graph not simply be 3-colourable, but 3-colourable in enough ways that there is no restriction to how any pair of nonadjacent vertices may be coloured: can they be coloured the same by some valid 3-colouring, and can they be coloured differently? This interesting combinatorial question can be rephrased in the following equally interesting way: is it true that every valid colouring of two vertices extends to a full 3-colouring? In the present article we look at a range of constraint problems where the question of satisfaction is replaced by the question as to whether there is a sufficiently flexible array of solutions to universally achieve various separation and identification properties on variables. We group this general notion loosely under the name flexible satisfaction, with specific kinds of flexible satisfaction to be given more precise definition. The idea appears natural enough in its own right, and so it is perhaps not surprising that it emerges from several different investigations of study. For example, there has been extensive exploration of the emergence of computationally challenging instances in randomly generated instances of constraint problems; see Hayes [16] or Monasson et al. [32] for example. Such work points toward the emergence of hidden constraints being intimately related to computational difficulty; see Beacham and Culberson [3] or Culberson and Gent [8] for example. These hidden 2010 Mathematics Subject Classification. Primary: 68Q17, 20M07; Secondary: 03C05, 08B99,