A Relaxation of a Semiring Constraint Satisfaction Problem Using Combined Semirings

The Semiring Constraint Satisfaction Problem (SCSP) framework is a popular approach for the representation of partial constraint satisfaction problems. In this framework preferences (semiring values) can be associated with tuples of values of the variable domains. Bistarelli et al. [1] define an abstract solution to a SCSP which consists of the best set of solution tuples for the variables in the problem. Sometimes this abstract solution may not be good enough, and in this case we want to change the constraints so that we solve a problem that is slightly di↵erent from the original problem but has an acceptable solution. In [2] we propose a relaxation of a SCSP where we define a measure of distance (a semiring value from a second semiring) between the original SCSP and a relaxed SCSP. In this paper we show how the two semirings can be combined into a single semiring. This combined semiring structure will allow us to use existing tools for SCSPs to solve Combined Semiring Relaxations of SCSPs. At this stage our work is preliminary and needs further investigation to develop into a useful algorithm.