[Paper Review] On the CNF encoding of cardinality constraints and beyond

Olivier Bailleux|arXiv (Cornell University)|Dec 17, 2010

Constraint Satisfaction and Optimization7 references3 citations

TL;DR

This paper provides a comprehensive survey of CNF encodings for Boolean cardinality constraints, evaluating techniques based on binary and unary bit counters and pseudo-Boolean encodings. It introduces a framework for analyzing encodings in terms of PAC (propagating arc consistency) and PIC (propagating inconsistency) properties, showing that unary-based encodings enable stronger unit propagation inference, and proposes informed unit propagation to enhance binary-based encodings, making them PIC-compliant.

ABSTRACT

In this report, we propose a quick survey of the currently known techniques for encoding a Boolean cardinality constraint into a CNF formula, and we discuss about the relevance of these encodings. We also propose models to facilitate analysis and design of CNF encodings for Boolean constraints.

Motivation & Objective

• To survey and evaluate existing techniques for encoding Boolean cardinality constraints into CNF formulas.
• To analyze the effectiveness of different encodings in supporting unit propagation for constraint reasoning.
• To propose a formal framework for assessing encodings based on PAC and PIC properties, which ensure strong consistency propagation.
• To investigate how unary versus binary representations of counter outputs affect inference power during unit propagation.
• To explore the potential of informed unit propagation to enhance binary-based encodings, making them PIC-compliant without increasing formula size.

Proposed method

• Classifies existing CNF encodings into two main categories: bit-counter-based (binary/unary) and pseudo-Boolean-based (e.g., BDD, LPW, GPW).
• Analyzes the Tseitin transformation of circuits combining bit counters and comparators to generate CNF formulas.
• Uses unary bit counters to achieve monotonicity in cardinality functions, enabling stronger unit propagation deductions.
• Introduces the concept of 'informed unit propagation' where unassigned variables are initially assumed to be false, allowing unit propagation to detect inconsistencies earlier.
• Constructs monotone Boolean circuits from CNF formulas to formally prove that unit propagation on a formula corresponds to monotone computation.
• Compares encodings using asymptotic complexity (e.g., Θ(n²), Θ(nk), Θ(n log²n)) and propagation properties to evaluate efficiency and inference power.

Experimental results

Research questions

• RQ1 What are the key differences in inference power between binary and unary bit counter encodings for cardinality constraints?
• RQ2 Why do binary-based encodings fail to achieve PAC or PIC properties, and how can this limitation be overcome?
• RQ3 Can informed unit propagation rules transform non-PIC binary encodings into PIC-compliant ones without increasing formula size?
• RQ4 How do the asymptotic complexities and propagation behaviors of BDD, LPW, and GPW encodings compare when applied to cardinality constraints?
• RQ5 Is there a compelling reason to prefer SAT solvers over native pseudo-Boolean solvers when both clauses and cardinality constraints are present in a problem?

Key findings

• Unary-based encodings achieve PAC and PIC properties because the unary representation ensures monotonicity in cardinality functions, enabling stronger unit propagation deductions.
• Binary-based encodings are neither PAC nor PIC because the non-monotonicity of binary bit functions prevents unit propagation from inferring cardinality constraints when variables are unassigned.
• The proposed informed unit propagation rule—assuming unassigned variables are initially false—can transform binary-based encodings into PIC-compliant ones, significantly improving inconsistency detection.
• The BDD-based encoding for cardinality constraints produces a formula of size O(nk), which is competitive with other encodings and maintains PAC properties.
• The LPW encoding for cardinality constraints degenerates into the unary counter encoding from [1], resulting in a Θ(nk) formula size, which is less efficient than other alternatives.
• A monotone Boolean circuit of size O(m) can simulate the deductions of unit propagation on a CNF formula of size m, proving that unit propagation corresponds to monotone computation.