Unified Framework for Semiring-Based Arc Consistency and Relaxation Labeling

Abstract

Constraint Satisfaction Problem (CSP), including its soft modifications, is ubiquitous in artificial intelligence and related fields. In computer vision and pattern recognition, the crisp CSP is more known as the consistent labeling problem and certain soft CSPs as certain inference problems in Markov Random Fields. Many soft CSPs can be seen as special cases of the semiring-based CSP (SCSP), using two abstract operations that form a semiring. A fundamental concept to tackle the CSP, as well as the SCSPs with idempotent semiring multiplication, are arc consistency algorithms, also known as relaxation labeling. Attempts have been made to generalize arc consistency for soft CSPs with non-idempotent semiring multiplication. We achieve such generalization by generalizing max-sum diffusion of Kovalevsky and Koval, used to decrease Schlesinger’s upper bound on the max-sum CSP. We formulate the proposed generalized arc consistency in the semiring framework. Newly, we introduce sum-product arc consistency and give its relation to max-sum arc consistency and optimal max-sum arc consistency.