Yet Another Comparison of SAT Encodings for the At-Most-K Constraint

This paper presents a comparison of SAT encodings for the at-most-k constraint, which is commonly used in combinatorial problems. Previous experiments have shown that the sequential-counter encoding is competitive for k > 1. The parallel-counter encoding is more compact than the binary-adder encoding but has been excluded due to its inability to enforce arc consistency through unit propagation. This paper introduces an experiment that demonstrates the remarkable performance of the binary-adder encoding for the at-most-k constraint. The experiment evaluates different encodings using the pigeonhole problem as a benchmark. The goal of this problem is to assign P pigeons to H holes, with each hole capable of holding up to K pigeons. The encodings are tested based on their CPU time using the Maple SAT solver. Table 1 provides a comparison of encodings for the at-most-one constraint in terms of CPU time and shows that some instances are small and unsatisfiable while others are large and satisfiable. Table 2 further compares encodings for ≤k (where k > 1) based on CPU time and reveals that the binary-adder encoding consistently performs better than both sequential counter and parallel counter encodings across various instances. The use of complete adders and enforcement of cardinality on all new variables contribute to its superior speed. In conclusion, this experiment demonstrates that the binary adder encoding exhibits astounding performance for solving combinatorial problems with the at most k constraint compared to other encodings such as sequential counter and parallel counter.

At-Most-K Constraint: A Comparison of SAT Encodings

Combinatorial problems are often solved using the at-most-k constraint, which limits the number of items that can be selected from a set. Previous experiments have shown that the sequential counter encoding is competitive for k > 1, while the parallel counter encoding is more compact but has been excluded due to its inability to enforce arc consistency through unit propagation. This paper introduces an experiment that demonstrates the remarkable performance of the binary adder encoding for this type of constraint.

The Pigeonhole Problem and Benchmarking

The experiment evaluates different encodings using the pigeonhole problem as a benchmark. The goal of this problem is to assign P pigeons to H holes, with each hole capable of holding up to K pigeons. The encodings are tested based on their CPU time using Maple SAT solver.

Table 1: At Most One Constraint Performance

Table 1 provides a comparison of encodings for the at-most-one constraint in terms of CPU time and shows that some instances are small and unsatisfiable while others are large and satisfiable.

Table 2: ≤k (where k > 1) Performance

Table 2 further compares encodings for ≤k (where k > 1) based on CPU time and reveals that the binary adder encoding consistently performs better than both sequential counter and parallel counter encodings across various instances. The use of complete adders and enforcement of cardinality on all new variables contribute to its superior speed.

Conclusion

In conclusion, this experiment demonstrates that binary adder encoding exhibits astounding performance for solving combinatorial problems with at most k constraints compared to other encodings such as sequential counter and parallel counter.