Bounds Arc Consistency in WCSPs

• Bounds Arc Consistency (BAC) is a local consistency method for WCSPs that prunes only domain bounds based on minimal cost supports.
• BAC employs accumulators and a work queue to update boundary costs efficiently, yielding significant time and memory savings.
• Empirical evaluations show BAC outperforms traditional soft arc consistency methods in scheduling and bioinformatics applications.

Bounds Arc Consistency (BAC) is a local consistency property tailored for Weighted Constraint Satisfaction Problems (WCSPs) with very large finite domains. BAC generalizes the concept of bounds consistency from crisp CSPs to the weighted (soft) setting. Instead of requiring full arc consistency across all domain values, BAC restricts consistency checks to domain bounds, providing substantial computational savings in both time and space, particularly in domains with high cardinality. BAC is implemented by testing and, if necessary, pruning only the minimal and maximal values of each variable’s domain based on the costs imposed by all relevant cost functions, thus never introducing potentially large unary tables. This property is confluent, ensuring that the resulting pruned network is unique, and it strictly dominates bounds consistency propagated over reified crisp encodings. BAC has demonstrated empirical superiority to established soft arc consistency methods, especially in large-scale scheduling and bioinformatics applications (Zytnicki et al., 2014).

1. Formal Framework and Definition

A WCSP is defined by a set of variables $X=\{x_1,\dots,x_n\}$, each with an ordered finite domain $D(x_i)$, and a set of cost functions $W$. For each variable $x_i$, the bounds $\inf(x_i)=\min D(x_i)$ and $\sup(x_i)=\max D(x_i)$ are defined. Every cost function $w_S \in W$ maps tuples $t$ (of assignments on $S \subseteq X$) to an integer in $[0,k]$, where $D(x_i)$0 is a threshold (possibly $D(x_i)$1) indicating a forbidden cost. Costs are combined using capped addition: $D(x_i)$2. The cost of a complete assignment $D(x_i)$3 is $D(x_i)$4.

A variable $D(x_i)$5 is bounds arc consistent if, for every $D(x_i)$6 with $D(x_i)$7, both bounds ($D(x_i)$8, $D(x_i)$9) admit a "soft support." Defining

$W$0

both inequalities

$W$1

must hold. If either condition is violated, the associated bound is pruned. Supports are given by tuples achieving the minimal cost.

2. Algorithmic Enforcement

The BAC enforcement procedure is structured around updating and checking only the boundary values:

• For each variable $W$2, maintain two accumulators: $W$3 and $W$4, capturing the minimal cost sums associated with respective bounds.
• Two corresponding arrays $W$5 store the minimal cost contributions from each cost function for rapid recomputation.
• A work queue tracks affected variables requiring bound re-examination.
• Iteratively, the procedure:
  • Pops a variable from the queue.
  • For each neighboring cost function, computes current minimum boundary costs.
  • Updates accumulators and, if necessary, deletes infeasible bounds and propagates through the queue.

This process uses $W$6 memory, dependent only on the number of variables and cost functions and not domain size.

3. Complexity Analysis

In the general case (arity $W$7), BAC can be enforced in $W$8 time and $W$9 space, where $x_i$0 is the number of cost functions and $x_i$1. For binary WCSPs ($x_i$2), this specializes to $x_i$3 time. Standard soft arc consistency (e.g., AC*) typically demands $x_i$4 time and $x_i$5 space. Thus, BAC achieves approximately a $x_i$6-fold speedup and significantly reduced memory footprint in large-domain settings. BAC enforcement is confluent and produces a unique pruned network (Zytnicki et al., 2014).

4. Cost Function Semantics and Acceleration

Performance bottlenecks in BAC stem from the need to recompute minimal supporting costs at domain bounds. These calculations can be significantly optimized for special classes of cost functions:

• Semi-Convex or Functional: If every binary cost is semi-convex (the set of high-cost values forms an interval) or functional (every $x_i$7 value has at most one cost-0 value in $x_i$8), then the minimal supporting value for each boundary can be found in $x_i$9, reducing BAC’s complexity to $\inf(x_i)=\min D(x_i)$0 overall.
• Convex or Monotonic: For convex or monotonic binary cost functions, the global minimum across all possible boundary assignments occurs at the domain extremes and can be found in $\inf(x_i)=\min D(x_i)$1, yielding an overall complexity of $\inf(x_i)=\min D(x_i)$2.

This semantic exploitation substantially accelerates BAC on structured constraint problems.

5. Comparison to Crisp Reified Bounds Consistency

Reification transforms each weighted cost function into a crisp constraint augmented by an integer cost variable, subject to a global constraint on their sum. Petit et al. (2000) formalized this approach. Zytnicki et al. demonstrate that BAC strictly dominates bounds consistency on any reified (crisp) encoding, i.e., BAC prunes every bound that would be deleted by this reified approach and may identify inconsistencies earlier (Zytnicki et al., 2014).

6. Empirical Evaluation on Real-World Problems

BAC has been empirically evaluated on large-scale satellite scheduling and RNA gene localization problems:

• In satellite scheduling, with task domains of size $\inf(x_i)=\min D(x_i)$3, BAC augmented with Ø-IC (notated BAC$\inf(x_i)=\min D(x_i)$4) is up to 35× faster than EDAC* while incurring only 25% more backtracks. Reified bounds consistency is observed to be 2–3× slower than BAC$\inf(x_i)=\min D(x_i)$5 for larger problem instances.
• In genome-scale RNA gene localization, with up to 22 variables and domain sizes reaching $\inf(x_i)=\min D(x_i)$6, BAC solves instances in less than three seconds using $\inf(x_i)=\min D(x_i)$7 space, whereas AC* exhausts available memory for $\inf(x_i)=\min D(x_i)$8. On the largest domains, BAC achieves speedups of $\inf(x_i)=\min D(x_i)$9 over AC*, with nearly identical numbers of backtracks, indicating that bounds-only filtering is typically sufficient to avoid infeasible regions.

Empirical findings establish the efficacy and scalability of BAC for large-domain WCSPs.

7. Summary and Applications

Bounds Arc Consistency is an efficient, domain-scaling local consistency for WCSPs, featuring $\sup(x_i)=\max D(x_i)$0 memory use, favorable worst-case enforcement times, and confluence. It strictly outperforms bounds consistency on reified crisp encodings and provides drastic performance improvements over classical soft arc consistency methods in practical settings such as temporal scheduling and genomic sequence analysis (Zytnicki et al., 2014).

A plausible implication is that BAC represents a key algorithmic advancement for constraint-based modeling in large, soft-constraint optimization domains, particularly where memory constraints and scalability are paramount.