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.