Using Pivot Consistency to Decompose and Solve Functional CSPs

TL;DR

This paper introduces a new decomposition method for functional constraint satisfaction problems that combines semantic and structural properties, utilizing pivot consistency to efficiently find solutions under certain conditions.

Contribution

It proposes a novel decomposition approach leveraging both semantic and structural properties, introducing pivot consistency as a weak form of path consistency with lower complexity.

Findings

Pivot consistency can be achieved in O(n^2d^2) complexity.

Consistent instantiations of the root set can be linearly extended to solutions.

The method guarantees solution existence under specific conditions.

Abstract

Many studies have been carried out in order to increase the search efficiency of constraint satisfaction problems; among them, some make use of structural properties of the constraint network; others take into account semantic properties of the constraints, generally assuming that all the constraints possess the given property. In this paper, we propose a new decomposition method benefiting from both semantic properties of functional constraints (not bijective constraints) and structural properties of the network; furthermore, not all the constraints need to be functional. We show that under some conditions, the existence of solutions can be guaranteed. We first characterize a particular subset of the variables, which we name a root set. We then introduce pivot consistency, a new local consistency which is a weak form of path consistency and can be achieved in O(n^2d^2) complexity (instead of O(n^3d^3) for path consistency), and we present associated properties; in particular, we show that any consistent instantiation of the root set can be linearly extended to a solution, which leads to the presentation of the aforementioned new method for solving by decomposing functional CSPs.