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Overall, one would naturally think that with n different nodes, and for x(1) for example representing node 1, it would be like:

x(1)+x(2)+x(3)...+x(n) <= k

This would mean that for every possible solution of k-vertex-cover, the maximum amount of literals being true has to be lower or equal to k. So much for understanding.

The problem however is, that a logic formula is needed for description of the constraint and logic doesn't have addition or comparison.

My only idea up to now is that I could use the logic formula of Adders and Comparators to simulate an addition and a comparison in logic, but that only works out for some solutions. I need a general logic formula. Does anyone have another idea or maybe a simpler method which could be used in general?

The reduction has to be done in polynomial time.

Archaeopteryx
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In "Efficient CNF encoding of Boolean cardinality constraints" the paper's authors offer a way of encoding at least and at most cardinality constraints using unary values. An advantage of this encoding over chains of adders and a comparator operating on a binary constraint is the immediate generation of conflict clauses when the constraint is violated during search and the immediate generation of unit clauses when the assignment of more of the constrained variables would exceed the constraint. This plays into the strength of modern DPLL-based SAT solvers optimized for speedy propagation of unit clauses. Though fairly complex in implementation the encoding is a nice compromise between the hideously inefficient naive encoding and the simple but search-inefficient adders+comparator encoding.

Kyle Jones
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