The factor graph of an instance of a constraint satisfaction problem (CSP) is the bipartite graph indicating which variables appear in each constraint. An instance of the CSP is given by the factor graph together with a list of which predicate is applied for each constraint. We establish that most Max-CSPs remain as hard to approximate as in the general case even when the factor graph is fixed (depending only on the size of the instance) and known in advance.

Examples of results obtained for this restricted setting are:

- Optimal inapproximability for Max-3-Lin and Max-3-Sat (Håstad, JACM 2001)
- Approximation resistance for predicates supporting pairwise independent subgroups (Chan, JACM 2016)
- Hardness of the “(2+ε)-Sat” problem and other Promise CSPs (Austrin et al., SICOMP 2017)

The main technical tool used to establish these results is a new way of folding the long code which we call “functional folding”.

Join work with Jonah Brown-Cohen and Johan Håstad.