  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 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
poly(max $|x_i|$) 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 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.
  Drucker (2012) 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 (1983) for 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.
