Circuit Complexity and Decompositions of Global Constraints

TL;DR

This paper explores how circuit complexity theory can analyze the decompositions of global constraints in constraint programming, establishing a link between circuit size and the feasibility of polynomial decompositions.

Contribution

It introduces a novel approach connecting circuit complexity with the decomposition of global constraints, providing lower bounds for certain propagators.

Findings

Polynomial size decompositions exist iff the propagator can be computed by a polynomial size monotone Boolean circuit.

Lower bounds on monotone circuit size imply lower bounds on global constraint decompositions.

No polynomial-sized decomposition exists for the domain consistency propagator of the ALLDIFFERENT constraint.

Abstract

We show that tools from circuit complexity can be used to study decompositions of global constraints. In particular, we study decompositions of global constraints into conjunctive normal form with the property that unit propagation on the decomposition enforces the same level of consistency as a specialized propagation algorithm. We prove that a constraint propagator has a a polynomial size decomposition if and only if it can be computed by a polynomial size monotone Boolean circuit. Lower bounds on the size of monotone Boolean circuits thus translate to lower bounds on the size of decompositions of global constraints. For instance, we prove that there is no polynomial sized decomposition of the domain consistency propagator for the ALLDIFFERENT constraint.