Drucker (2012) proved the following result: Unless the unlikely complexity-theoretic collapse coNP is in 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 simple proof of this result.

An AND-compression is a deterministic polynomial-time algorithm that maps a set of SAT-instances x1xt to a single SAT-instance y of size poly(max xi ) such that y is satisfiable if and only if all xi 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 "xi is satisfiable". Drucker's result complements the result by Bodlaender et al. (2009) and Fortnow and Santhanam (2010), who proved the analogous statement for OR-compressions, and Drucker's proof not only subsumes that result but also extends it to randomized compression algorithms that are allowed to have a certain probability of failure.

The overall structure of our proof is similar to the arguments of Ko (1983) for P-selective sets, which use the fact that tournaments have dominating sets of logarithmic size. We generalize this fact to hypergraph tournaments. For the information-theoretic part of the proof, we consider a natural generalization of the average noise sensitivity of a Boolean function, which is bounded for compressive maps. We prove this with mechanical calculations that involve the Kullback-Leibler divergence