Tractability and Decompositions of Global Cost Functions

TL;DR

This paper investigates the conditions under which applying Equivalent Preserving Transformations (EPTs) to global cost functions preserves tractability, introducing methods for efficient computation and decomposition to enhance local consistency enforcement.

Contribution

It establishes the relationship between EPT scope size and tractability preservation, and proposes dynamic programming algorithms for computing EPTs on certain cost functions.

Findings

EPTs preserve tractability when scope size r=0

Decomposition into Berge acyclic networks enables efficient local consistency enforcement

Experimental results confirm the efficiency of the proposed methods

Abstract

Enforcing local consistencies in cost function networks is performed by applying so-called Equivalent Preserving Transformations (EPTs) to the cost functions. As EPTs transform the cost functions, they may break the property that was making local consistency enforcement tractable on a global cost function. A global cost function is called tractable projection-safe when applying an EPT to it is tractable and does not break the tractability property. In this paper, we prove that depending on the size r of the smallest scopes used for performing EPTs, the tractability of global cost functions can be preserved (r = 0) or destroyed (r > 1). When r = 1, the answer is indefinite. We show that on a large family of cost functions, EPTs can be computed via dynamic programming-based algorithms, leading to tractable projection-safety. We also show that when a global cost function can be decomposed into a Berge acyclic network of bounded arity cost functions, soft local consistencies such as soft Directed or Virtual Arc Consistency can directly emulate dynamic programming. These different approaches to decomposable cost functions are then embedded in a solver for extensive experiments that confirm the feasibility and efficiency of our proposal.