  The factor graph of an instance of a symmetric constraint satisfaction
problem on n Boolean variables and m constraints (CSPs such as k-SAT, k-AND,
k-LIN) is a bipartite graph describing which variables appear in which
constraints. The factor graph describes the instance up to the polarity of the
variables, and hence there are up to 2km instances of the CSP that share the
same factor graph. It is well known that factor graphs with certain structural
properties make the underlying CSP easier to either solve exactly (e.g., for
tree structures) or approximately (e.g., for planar structures). We are
interested in the following question: is there a factor graph for which if one
can solve every instance of the CSP with this particular factor graph, then one
can solve every instance of the CSP regardless of the factor graph (and
similarly, for approximation)? We call such a factor graph universal. As one
needs different factor graphs for different values of n and m, this gives rise
to the notion of a family of universal factor graphs. We initiate a systematic
study of universal factor graphs, and present some results for max-kSAT. Our
work has connections with the notion of preprocessing as previously studied for
closest codeword and closest lattice-vector problems, with proofs for the PCP
theorem, and with tests for the long code. Many questions remain open.
