Reduce Vertex cover to SAT

Problem Detail: 

I need to reduce the vertex cover problem to a SAT problem, or rather tell whether a vertex cover of size k exists for a given graph, after solving with a SAT solver. I know how to reduce a 3-SAT problem to vertex cover problem, by constructing the subgraphs for each variable (x, !x) and for each clause (a triable). But I am not getting,how to do other way round?

I was thinking of first forming a DNF ,with electing k vertices at first and then convert it to a CNF, by enumerating all clauses. Is there any other method?

Asked By : Amrith Krishna

Answered By : Niklas B.

Let's introduce a variable $x_i$ for every node $i$, representing the condition that the node is part of the vertex cover. Then for every edge $\{v,w\}$ we introduce the clause $x_v \lor x_w$.

We have the additional condition $x_1 + x_2 + \ldots \ +\ x_n \leq k$. We can combine an addition circuit to compute the bits of the sum with a comparator circuit to enforce the inequality. The CNF of the resulting circuit has only polynomially many clauses.

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Question Source : http://cs.stackexchange.com/questions/22426