  A parameterized problem consists of a classical problem and an additional
component, the so-called parameter. This point of view allows a formal
definition of preprocessing: Given a parameterized instance (I,k), a polynomial
kernelization computes an equivalent instance (I',k') of size and parameter
bounded by a polynomial in k. We give a complete classification of Min Ones
Constraint Satisfaction problems, i.e., Min Ones SAT(\Gamma), with respect to
admitting or not admitting a polynomial kernelization (unless NP \subseteq
coNP/poly). For this we introduce the notion of mergeability. If all relations
of the constraint language \Gamma are mergeable, then a new variant of
sunflower kernelization applies, based on non-zero-closed cores. We obtain a
kernel with O(k^{d+1}) variables and polynomial total size, where d is the
maximum arity of a constraint in \Gamma, comparing nicely with the bound of
O(k^{d-1}) vertices for the less general and arguably simpler d-Hitting Set
problem. Otherwise, any relation in \Gamma that is not mergeable permits us to
construct a log-cost selection formula, i.e., an n-ary selection formula with
O(log n) true local variables. From this we can construct our lower bound using
recent results by Bodlaender et al. as well as Fortnow and Santhanam, proving
that there is no polynomial kernelization, unless NP \subseteq coNP/poly and
the polynomial hierarchy collapses to the third level.
