A Maximal Tractable Class of Soft Constraints
  Many researchers in artificial intelligence are beginning to explore the use
of soft constraints to express a set of (possibly conflicting) problem
requirements. A soft constraint is a function defined on a collection of
variables which associates some measure of desirability with each possible
combination of values for those variables. However, the crucial question of the
computational complexity of finding the optimal solution to a collection of
soft constraints has so far received very little attention. In this paper we
identify a class of soft binary constraints for which the problem of finding
the optimal solution is tractable. In other words, we show that for any given
set of such constraints, there exists a polynomial time algorithm to determine
the assignment having the best overall combined measure of desirability. This
tractable class includes many commonly-occurring soft constraints, such as 'as
near as possible' or 'as soon as possible after', as well as crisp constraints
such as 'greater than'. Finally, we show that this tractable class is maximal,
in the sense that adding any other form of soft binary constraint which is not
in the class gives rise to a class of problems which is NP-hard.
