Arc Consistency and Friends

A natural and established way to restrict the constraint satisfaction problem is to fix the relations that can be used to pose constraints; such a family of relations is called a constraint language. In this article, we study arc consistency, a heavily investigated inference method, and three extensions thereof from the perspective of constraint languages. We conduct a comparison of the studied methods on the basis of which constraint languages they solve, and we present new polynomial-time tractability results for singleton arc consistency, the most powerful method studied.

💡 Research Summary

The paper “Arc Consistency and Friends” investigates four well‑known consistency‑based inference methods for the constraint satisfaction problem (CSP) – Arc Consistency (AC), Look‑Ahead Arc Consistency (LAAC), Peek Arc Consistency (PAC) and Singleton Arc Consistency (SAC) – from the viewpoint of constraint languages, i.e., fixed right‑hand side structures (templates) B. The authors first formalize CSP as a homomorphism problem: given two relational structures A and B over the same signature, decide whether there exists a homomorphism A → B. Fixing B yields the family CSP(B), which captures many classic problems such as Boolean SAT, graph homomorphism, and algebraic equation solving.

For each algorithm the paper provides a uniform homomorphism‑based consistency condition. AC is shown to be equivalent to the existence of a homomorphism A → ℘(B), where ℘(B) is the structure of all non‑empty subsets of B equipped with the lifted relations. Consequently, AC solves CSP(B) exactly when there is a homomorphism ℘(B) → B. LAAC, which builds a solution variable by variable while repeatedly invoking AC as a filter, is characterized by the existence of a binary homomorphism ℘(B) × B → B satisfying l({b},b′)=b. PAC, which “peeks” at each variable by testing every possible value with a single AC call, is captured by a similar algebraic condition. The authors prove that these three methods form a strict hierarchy: AC ⊆ LAAC ⊆ PAC ⊆ SAC, meaning that any language solvable by a weaker method is automatically solvable by the stronger ones, but not vice‑versa.

The central technical contribution concerns Singleton Arc Consistency. SAC runs AC on each singleton restriction (A,{a}) vs. (B,{b}) and removes values that cause a failure. Previously, the best known worst‑case bound for SAC on tractable languages was cubic in the size of B. The authors establish new polynomial‑time tractability results that improve this to quadratic time for two broad classes of languages:

1. Languages with a 2‑semilattice polymorphism – i.e., a binary operation ∧ that is associative, commutative, idempotent, and preserves every relation of B. The paper proves that for any finite B admitting such an operation, there exists a homomorphism ℘(B) → B, implying that SAC decides CSP(B) completely. This yields a quadratic‑time algorithm for all such languages.

2. Languages with a majority polymorphism – a ternary operation maj(x,y,z) that returns the majority value among its arguments and preserves all relations of B. The authors show that the presence of a majority polymorphism also guarantees a homomorphism ℘(B) → B, and therefore SAC is a complete decision procedure. Since majority polymorphisms already characterize languages where 3‑consistency implies global consistency, this result connects two previously separate strands of CSP theory.

Both results are significant because they provide alternative, algorithmically simple proofs of tractability for large, well‑studied language families, and they demonstrate that SAC can be implemented in O(|B|²) time rather than the previously assumed O(|B|³). Moreover, the algebraic proofs reveal structural properties of the underlying templates that are invisible to earlier combinatorial arguments.

The paper also discusses the practical implications: AC, LAAC and PAC are already standard preprocessing steps in many CSP solvers; the authors’ algebraic characterizations allow practitioners to predict, based solely on the polymorphisms of the template, whether these preprocessing steps will be sufficient to solve the instance completely. SAC, being strictly more powerful yet still polynomial, offers a viable option for harder instances where AC‑based filtering fails.

In conclusion, “Arc Consistency and Friends” unifies the analysis of several consistency algorithms under the algebraic framework of polymorphisms, establishes a clear hierarchy among them, and delivers new tractability results for SAC on languages with 2‑semilattice or majority polymorphisms, improving the known runtime bounds. The work opens avenues for further exploration of other polymorphism classes (e.g., Maltsev, near‑unanimity) and for extending the hierarchy to broader families of constraint languages.