Path-Consistency Algorithm Overview

• Path-Consistency Algorithm (PCA) is a local consistency method that ensures every pair of variable assignments can be consistently extended to a third variable in binary constraint networks.
• Directional variants like DPC and DPC* leverage an imposed variable ordering to enhance efficiency and decisively solve CSPs, particularly in majority-closed constraint languages.
• Empirical studies show DPC* achieves significant speedups over traditional algorithms, providing scalable, backtrack-free solving in applications such as temporal, spatial, and scene labeling.

The path-consistency algorithm (PCA) is a fundamental local consistency technique in constraint satisfaction, particularly within binary constraint networks (BCNs). Path-consistency (PC) refines the global structure of a constraint network by ensuring that the allowed assignments for every pair of variables can be consistently extended to every third variable. Directional path-consistency (DPC) is a refinement of PC that leverages an imposed variable ordering to enable more efficient enforcement. The DPC family of algorithms, including DPC* for majority-closed constraint languages, play a decisive role in the tractability of numerous CSP subclasses and underlie efficient, backtrack-free solving in domains such as geometric reasoning, temporal reasoning, and scene labeling (Kong et al., 2017).

1. Formal Definition and Classical Path-Consistency

Let $\mathcal{N} = (V, D, C)$ be a binary constraint network (BCN) with variables $V = \{v_1, ..., v_n\}$, finite domains $D = \{D_1, ..., D_n\}$, and binary constraints $C = \{(v_i, v_j, R_{ij})\}$. The constraint graph $G = (V, E)$ contains an edge $\{v_i, v_j\}$ iff $R_{ij} \in C$.

A network is path-consistent (PC) if, for every triple $(v_i, v_k, v_j)$ where $\{v_i, v_j\} \in E$, and for every pair $(a,c) \in R_{ij}$, there exists $V = \{v_1, ..., v_n\}$0 such that $V = \{v_1, ..., v_n\}$1 and $V = \{v_1, ..., v_n\}$2. PC is enforced iteratively, updating relations to maintain this property. The canonical PC-3 algorithm maintains a queue of such triples, propagating constraint refinements until closure or inconsistency. The worst-case time complexity is $V = \{v_1, ..., v_n\}$3 for $V = \{v_1, ..., v_n\}$4 variables, with $V = \{v_1, ..., v_n\}$5 (Kong et al., 2017).

2. Directional Path-Consistency (DPC) and Enforcement

Directional path-consistency (DPC), as proposed by Dechter and Pearl, exploits a total ordering $V = \{v_1, ..., v_n\}$6 on $V = \{v_1, ..., v_n\}$7 (e.g., $V = \{v_1, ..., v_n\}$8). Three notions are introduced:

• Directionally arc-consistent (DAC): For every arc $V = \{v_1, ..., v_n\}$9 with $D = \{D_1, ..., D_n\}$0, every $D = \{D_1, ..., D_n\}$1 has some $D = \{D_1, ..., D_n\}$2 with $D = \{D_1, ..., D_n\}$3.
• DPC: For every edge $D = \{D_1, ..., D_n\}$4 and every $D = \{D_1, ..., D_n\}$5 with $D = \{D_1, ..., D_n\}$6 and $D = \{D_1, ..., D_n\}$7, the triple $D = \{D_1, ..., D_n\}$8 is PC.
• Strong DPC: Both DAC and DPC hold throughout the network.

Strong DPC is enforceable in a single backward pass via Algorithm 1 (strong-DPC): for $D = \{D_1, ..., D_n\}$9 down to $C = \{(v_i, v_j, R_{ij})\}$0, first enforce DAC for each $C = \{(v_i, v_j, R_{ij})\}$1, then, for each $C = \{(v_i, v_j, R_{ij})\}$2 with $C = \{(v_i, v_j, R_{ij})\}$3, enforce PC via the update $C = \{(v_i, v_j, R_{ij})\}$4. The time complexity is $C = \{(v_i, v_j, R_{ij})\}$5, with $C = \{(v_i, v_j, R_{ij})\}$6 the induced width along ordering $C = \{(v_i, v_j, R_{ij})\}$7 (Kong et al., 2017).

3. Decidability Conditions: Variable Elimination Property and the Helly Property

DPC decides the CSP for a constraint language $C = \{(v_i, v_j, R_{ij})\}$8 if and only if $C = \{(v_i, v_j, R_{ij})\}$9 possesses the variable elimination property (VEP): for every variable $G = (V, E)$0, any solution of the projected network $G = (V, E)$1 (where $G = (V, E)$2 is eliminated) can be extended to a solution to the full network. The structural characterization is formalized as:

Theorem 3.1: For a complete set of binary relations $G = (V, E)$3, strong-DPC decides CSP$G = (V, E)$4 if and only if $G = (V, E)$5 has VEP (Kong et al., 2017).

VEP is linked to the Helly property for binary relations: for any collection $G = (V, E)$6, a common intersection exists for all $G = (V, E)$7 whenever every pairwise intersection is nonempty.

A key consequence is that no complete VEP constraint language admits domains of size greater than 2, constraining the scope of DPC's direct applicability (Kong et al., 2017).

4. The DPC* Algorithm for Majority-Closed Constraint Languages

Many tractable BCN classes, such as connected row-convex (CRC) and tree-preserving constraints, are majority-closed (admit a majority polymorphism) but do not satisfy VEP when $G = (V, E)$8. DPC* adapts the strong-DPC algorithm by weakening DAC enforcement:

Let $G = (V, E)$9 be a multi-sorted majority operation satisfying $\{v_i, v_j\}$0 $\{v_i, v_j\}$1. $\{v_i, v_j\}$2 is closed under $\{v_i, v_j\}$3 iff $\{v_i, v_j\}$4, $\{v_i, v_j\}$5. $\{v_i, v_j\}$6 is majority-closed if some $\{v_i, v_j\}$7 preserves every $\{v_i, v_j\}$8.

Algorithm 2 (DPC*):

For each variable in reverse order:

• If $\{v_i, v_j\}$9 has only one later neighbor $R_{ij} \in C$0, apply one-sided arc-consistency: $R_{ij} \in C$1.
• Otherwise, for each $R_{ij} \in C$2: $R_{ij} \in C$3.
• For each $R_{ij} \in C$4, $R_{ij} \in C$5: $R_{ij} \in C$6.

Theorem 4.1: If $R_{ij} \in C$7 is a complete majority-closed language, DPC* decides CSP$R_{ij} \in C$8 and produces a decomposable network: any partial solution can be extended to a global solution without backtracking (Kong et al., 2017).

5. Notable Majority-Closed Classes and Applications

Two prominent majority-closed BCN classes are:

• Connected Row-Convex (CRC) Constraints: Domains are chains (totally ordered sets), with allowed sets projecting to row-convex sets in the other domain. CRC constraints are employed in temporal reasoning, geometric reasoning, and logical filtering (Kong et al., 2017).
• Tree-Preserving Constraints: Domains are trees, and each $R_{ij} \in C$9 preserves connectivity of subtrees. These constraints are instrumental in scene labeling problems in vision, map labeling, and spatial reasoning (Kong et al., 2017).

Applications of these classes often require efficient, backtrack-free CSP solving, a property guaranteed in majority-closed languages via DPC*.

6. Empirical Comparison and Performance Analysis

Empirical evaluations compared DPC* with two state-of-the-art local consistency solvers:

• PC2001: An optimized PC algorithm [Bessière et al. ’05].
• SAC3-SDS: A singleton arc-consistency solver [Bessière–Debruyne ’11].

Tested on tree-preserving and random majority-closed networks (domain size $(v_i, v_k, v_j)$0, density $(v_i, v_k, v_j)$1, looseness $(v_i, v_k, v_j)$2), DPC* demonstrated the following results (Kong et al., 2017):

$(v_i, v_k, v_j)$3 (variables)	PC2001 time	SAC3-SDS time	DPC* time
20	76 s	52 s	16 s
60	211 s	120 s	28 s
120	323 s	230 s	65 s

DPC* scales linearly in $(v_i, v_k, v_j)$4, achieves $(v_i, v_k, v_j)$5–$(v_i, v_k, v_j)$6 speedups over PC2001 and $(v_i, v_k, v_j)$7–$(v_i, v_k, v_j)$8 over SAC3-SDS, and remains $(v_i, v_k, v_j)$9 in domain size while others scale as $\{v_i, v_j\} \in E$0–$\{v_i, v_j\} \in E$1. Performance is stable across varying density and looseness, attributed to single-pass elimination and majority-driven decomposability.

7. Summary and Context

The path-consistency algorithm and its directional variants, particularly DPC*, form an essential component for the efficient resolution of CSPs in both theoretical and applied settings. Decidability by DPC depends on the VEP, while majority-closed languages—encompassing key domains in temporal, spatial, and logical reasoning—are decisively solvable by the DPC* algorithm, yielding decomposability and empirical speed advantages. This situates DPC and DPC* as central algorithmic tools for constraint-based inference where structure and polymorphism properties can be exploited (Kong et al., 2017).