## Tractability and Decompositions of Global Cost Functions

- David Allouche, Christian Bessiere, +8 authors Yi Wu
- ArXiv
- 2015

- Published 2016 in Artif. Intell.

Graphical model processing is a central problem in artificial intelligence. The optimization of the combined cost of a network of local cost functions federates a variety of famous problems including CSP, SAT and Max-SAT but also optimization in stochastic variants such as Markov Random Fields and Bayesian networks. Exact solving methods for these problems typically include branch and bound and local inference-based bounds. In this paper we are interested in understanding when and how dynamic programming based optimization can be used to efficiently enforce soft local consistencies on Global Cost Functions, defined as parameterized families of cost functions of unbounded arity. Enforcing local consistencies in cost function networks is performed by applying so-called Equivalence Preserving Transformations (EPTs) to the cost functions. These EPTs may transform global cost functions and make them intractable to optimize. We identify as tractable projection-safe those global cost functions whose optimization is and remains tractable after applying the EPTs used for enforcing arc consistency. We also provide new classes of cost functions that are tractable projection-safe thanks to dynamic programming. We show that dynamic programming can either be directly used inside filtering algorithms, defining polynomially DAG-filterable cost functions, or emulated by arc consistency filtering on a Berge-acyclic network of bounded-arity cost functions, defining Berge-acyclic network-decomposable cost functions. We give examples of such cost functions and we provide a systematic way to define decompositions from existing decomposable global constraints. These two approaches to enforcing consistency in global cost functions are then embedded in a solver for extensive experiments that confirm the feasibility and efficiency of our proposal. ✩This paper is an extended version of [3], DOI 10.1016/j.artint.2016.06.005. Corresponding author Preprint submitted to Artificial Intelligence June 30, 2016

@article{Allouche2016TractabilitypreservingTO,
title={Tractability-preserving transformations of global cost functions},
author={David Allouche and Christian Bessiere and Patrice Boizumault and Simon de Givry and Patricia Gutierrez and Jimmy Ho-Man Lee and Ka Lun Leung and Samir Loudni and Jean-Philippe M{\'e}tivier and Thomas Schiex and Yi Wu},
journal={Artif. Intell.},
year={2016},
volume={238},
pages={166-189}
}