Soft constraint abstraction based on semiring homomorphism
  The semiring-based constraint satisfaction problems (semiring CSPs), proposed
by Bistarelli, Montanari and Rossi \cite{BMR97}, is a very general framework of
soft constraints. In this paper we propose an abstraction scheme for soft
constraints that uses semiring homomorphism. To find optimal solutions of the
concrete problem, the idea is, first working in the abstract problem and
finding its optimal solutions, then using them to solve the concrete problem.
  In particular, we show that a mapping preserves optimal solutions if and only
if it is an order-reflecting semiring homomorphism. Moreover, for a semiring
homomorphism $\alpha$ and a problem $P$ over $S$, if $t$ is optimal in
$\alpha(P)$, then there is an optimal solution $\bar{t}$ of $P$ such that
$\bar{t}$ has the same value as $t$ in $\alpha(P)$.
