Fields Summer School on Approximability of CSPs

These lectures will be about the approximability of constraint satisfaction problems (CSPs). Definition 1.1. Let D be a domain of cardinality q. We usually write D = {0,1,2, . . . , q − 1} and often call the elements of D labels. Let Γ be a nonempty finite set of nontrivial relations (predicates) over D, with each R ∈ Γ having arity ar(R) ≤ k. We usually write an r-ary relation as R : Dr → {0,1}. Together, D and Γ form an algorithmic problem called CSP(Γ). Definition 1.2. An instance P of CSP(Γ) over the n variables V consists of a multiset of m constraints. Each constraint C ∈ P is a pair (R,S), where R ∈ Γ and S (the scope of C) is list of ar(R) many distinct1 variables from V . We assume that each variable is in the scope of at least one constraint, so m ≥ n/k. The algorithmic goal, given an instance P , is to find an assignment for P which is as “good” as possible. Definition 1.3. An assignment for an instance P of CSP(Γ) is any mapping F : V → D. The assignment satisfies a constraint C = (R,S) if R(F(S))= 1, where the notation F(S) means (F(S1), . . . ,F(Sr)). The value of the assignment, ValP (F) ∈ [0,1], is the fraction of constraints it satisfies; this can be written ValP (F)= avg C=(R,S)∈P [R(F(S))].