Soft constraint abstraction based on semiring homomorphism

TL;DR

This paper introduces a novel abstraction scheme for soft constraints using semiring homomorphisms, enabling efficient solution transfer from an abstract to a concrete problem in semiring CSPs.

Contribution

It proposes a new semiring homomorphism-based abstraction method for soft constraints, ensuring preservation of optimal solutions between abstract and concrete problems.

Findings

A semiring homomorphism preserves optimal solutions if and only if it is order-reflecting.

Optimal solutions in the abstract problem correspond to optimal solutions in the concrete problem.

The method facilitates solving complex soft constraint problems via simpler abstract problems.

Abstract

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 $\alpha(P)$.