# AND-compression of NP-complete Problems: Streamlined Proof and Minor Observations

## Abstract

Drucker [8] proved the following result: Unless the unlikely complexity-theoretic collapse \(\mathsf {coNP}\subseteq \mathsf {NP/poly}\) occurs, there is no *AND-compression* for SAT. The result has implications for the compressibility and kernelizability of a whole range of NP-complete parameterized problems. We present a streamlined proof of Drucker’s theorem.

An AND-compression is a deterministic polynomial-time algorithm that maps a set of SAT-instances \(x_1,\dots ,x_t\) to a single SAT-instance \(y\) of size \(\mathrm{poly }(\max _i|x_i|)\) such that \(y\) is satisfiable if and only if all \(x_i\) are satisfiable. The “AND” in the name stems from the fact that the predicate “\(y\) is satisfiable” can be written as the AND of all predicates “\(x_i\) is satisfiable”. Drucker’s theorem complements the result by Bodlaender et al. [3] and Fortnow and Santhanam [10], who proved the analogous statement for OR-compressions, and Drucker’s proof not only subsumes their result but also extends it to *randomized* compression algorithms that are allowed to have a certain probability of failure.

Drucker [8] presented two proofs: The first uses information theory and the minimax theorem from game theory, and the second is an elementary, iterative proof that is not as general. In our proof, we realize the iterative structure as a generalization of the arguments of Ko [12] for \(\mathsf {P}\)-selective sets, which use the fact that tournaments have dominating sets of logarithmic size. We generalize this fact to hypergraph tournaments. Our proof achieves the full generality of Drucker’s theorem, avoids the minimax theorem, and restricts the use of information theory to a single, intuitive lemma about the average noise sensitivity of compressive maps. To prove this lemma, we use the same information-theoretic inequalities as Drucker.

## Notes

### Acknowledgments

I would like to thank Andrew Drucker, Martin Grohe, and others for encouraging me to pursue the publication of this manuscript, David Xiao for pointing out Theorem 5 to me, Andrew Drucker, Dániel Marx, and anonymous referees for comments on an earlier version of this paper, and Dieter van Melkebeek for some helpful discussions.

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