Arc-consistency and linear programming duality: an analysis of reduced cost based filtering

TL;DR

This paper explores how linear programming duality and reduced costs can efficiently achieve arc-consistency in constraint programming with costs, reducing the number of LP calls needed from exponential to linear in the number of variables.

Contribution

It introduces a novel analysis linking LP dual solutions to arc-consistency, providing a simple n-call algorithm for ideal LP formulations and extending previous work on satisfaction problems.

Findings

n dual solutions suffice for AC in ideal LP cases

A linear-time algorithm with n LP calls is proposed

Extended analysis from satisfaction to cost-based constraints

Abstract

In Constraint Programming (CP), achieving arc-consistency (AC) of a global constraint with costs consists in removing from the domains of the variables all the values that do not belong to any solution whose cost is below a fixed bound. We analyse how linear duality and reduced costs can be used to find all such inconsistent values. In particular, when the constraint has an ideal Linear Programming (LP) formulation, we show that n dual solutions are always enough to achieve AC (where n is the number of variables). This analysis leads to a simple algorithm with n calls to an LP solver to achieve AC, as opposed to the naive approach based on one call for each value of each domain. It extends the work presented in [German et al., 2017] for satisfaction problems and in [Claus et al., 2020] for the specific case of the minimum weighted alldifferent constraint. We propose some answers to the following questions: does there always exists a dual solution that can prove a value consistent/inconsistent ? given a dual solution, how do we know which values are proved consistent/inconsistent ? can we identify simple conditions for a family of dual solutions to ensure arc-consistency ?