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I have a simple looking question. What is the most efficient conversion of $\sum_{i=1}^n x_i = y$ to 3-sat? Here $x_i$ is either $1$ or $0$ and $y$ is some positive integer.

Can you do better than making a SATISFIABILITY instance with $\binom{n}{y}$ clauses, each of which is the conjunction of $y$ positive literals and $n-y$ negative literals and then just feeding the whole thing into the Tseitin transform?

Simd
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2 Answers2

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Many better techniques for enforcing cardinality constraints are described in this answer. For the special case $y=1$, see also Encoding 1-out-of-n constraint for SAT solvers. Read those links; they suggest more efficient conversions, though I'm not aware of any reason to expect that they are necessarily optimal, so they don't answer your question about the most efficient conversion.

Community
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D.W.
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  • Thank you. The " Encoding 1-out-of-n constraint" link looks very useful for that problem. I am still working through the other interesting links but I am not sure they fully answer my question. – Simd Sep 29 '13 at 14:42
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Hint: Use a circuit for computing $\sum_{i=1}^n x_i$. Don't forget you're allowed to add variables!

Yuval Filmus
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  • Would you mind expanding this answer a bit please? – Simd Sep 27 '13 at 18:03
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    Well, it's your exercise. You can look up the proof of Cook's theorem (along with the reduction from SAT to 3SAT), which explains how to convert a circuit to a SAT instance. – Yuval Filmus Sep 27 '13 at 21:16