Conditional Lexicographic Orders in Constraint Satisfaction Problems

The lexicographically-ordered CSP (“lexicographic CSP” for short) combines a simple representation of preferences with the feasibility constraints of ordinary CSPs. Preferences are defined by a total ordering across all assignments, such that a change in assignment to variable

k

is more important than any change in assignment to any variable that comes after it in the ordering. In this paper, we show how this representation can be extended to handle conditional preferences. This can be done in two ways. In the first, for each conditional preference relation, the parents have higher priority than the children in the original lexicographic ordering. In the second, the relation between parents and children need not correspond to the basic ordering of variables. For problems of the first type, any of the algorithms originally devised for ordinary lexicographic CSPs can also be used when some of the domain orderings are dependent on the assignments to “parent” variables. For problems of the second type, algorithms based on lexical orders can be used if the representation is augmented by variables and constraints that link preference orders to assignments. In addition, the branch-and-bound algorithm originally devised for ordinary lexicographic CSPs can be extended to handle CSPs with conditional domain orderings.