The complexity of constraint satisfaction: an algebraic approach

Dualities for Constraint Satisfaction Problems

“…The algebraic approach to constraint satisfaction (see, e.g., [10][11][12]44]) has proved to be extremely successful. It provides a convenient dual language to analyse CSPs, and, more importantly, allows one to use powerful machinery from universal algebra.…”

“…Schaefer's celebrated dichotomy theorem for Boolean CSP can be restated (see, e.g., [10,12,44]) as follows. For a Boolean core structure B, if B has a semilattice polymorphism, or a majority polymorphism, or the affine polymorphism, then CSP(B) is in PTIME.…”

“…It is known (see [10,11,44]) that if, for a core structure B, the variety var(A B ) admits type 1 then CSP(B) is NP-complete. Moreover, all core structures B that are known to give rise to NP-complete problems CSP(B) do satisfy this condition.…”

“…The obtained problem is denoted by CSP(B). Examples of such problems include various versions of k-Sat, Graph Colouring, and Systems of Equations (see [12,28,36,44]). Strong motivation for studying this framework was given in [22] where it was shown that such problems can be used in attempts to identify a largest subclass of NP that avoids problems of intermediate complexity.…”

“…The most advanced approaches use logic (e.g., [42]), combinatorics (e.g., [26,28,46]), universal algebra (e.g., [6,9,10,12,35,44]), or combinations of those (e.g., [2,7,16,22,48]). In this survey, we will discuss a combinatorial idea that has a bearing on all the above problems, and has strong links with the three approaches -the idea of homomorphism duality.…”