Independent set, vertex cover, k-coloring and clique admit CNF encodings with O(n^2/log n) clauses, and the disjoint-intervals encoding needs only O(n^2 log n) clauses.
Yet Another Comparison of SAT Encodings for the At-Most-K Constraint
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abstract
The at-most-k constraint is ubiquitous in combinatorial problems, and numerous SAT encodings are available for the constraint. Prior experiments have shown the competitiveness of the sequential-counter encoding for k $>$ 1, and have excluded the parallel-counter encoding, which is more compact that the binary-adder encoding, from consideration due to its incapability of enforcing arc consistency through unit propagation. This paper presents an experiment that shows astounding performance of the binary-adder encoding for the at-most-k constraint.
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Asymptotically Smaller Encodings for Graph Problems and Scheduling
Independent set, vertex cover, k-coloring and clique admit CNF encodings with O(n^2/log n) clauses, and the disjoint-intervals encoding needs only O(n^2 log n) clauses.