COLOURINGS OF HYPERGRAPHS, PERMUTATION GROUPS ...

2 B. LAROSE AND L. HADDAD
give various examples of both types of behaviour, and in [13] Haddad, Hell and
Mendelsohn investigate this problem further.
We now reformulate this decision problem in a more convenient form. Let A =
{1, . . . , h}.
• CSP (ρ G )
Input: a relational structure 〈X; µ〉 where µ is h-ary;
Question: is there a homomorphism from 〈X; µ〉 to 〈A; ρ G 〉
It is easy to see that in fact these problems are equivalent. Indeed, it is clear
that if the input structure for CSP (ρ G ) is not areflexive then there cannot be a
homomorphism. Furthermore, it is easy to verify that if a homomorphism exists,
and if both (x 1 , . . . , x h ) and (x π(1) , . . . , x π(h) ) are in µ then π ∈ G. Finally, we can
always add to µ all permutation of its tuples by permutations in G without changing
the set of solutions. Obviously all these steps can be done in time polynomial in
the size of the structure 〈X; µ〉 and thus we may safely restrict the inputs to the
problem CSP (ρ G ) (abbreviated from now as CSP (G)) to areflexive structures with
symmetry group G.
More generally, for any finite set of relations R = {ρ 1 , . . . , ρ k } on A where ρ i has
arity d i , let CSP (R) denote the following decision problem:
• CSP (R)
Input: a relational structure 〈X; µ 1 , . . . , µ k 〉 where µ i is d i -ary;
Question: is there a homomorphism from 〈X; µ 1 , . . . , µ k 〉 to 〈A; ρ 1 , . . . , ρ k 〉
In 1993, Feder and Vardi [11] conjectured that, depending on the constraint
relations, the problem CSP (ρ 1 , . . . , ρ k ) should be either in P or NP-complete.
This dichotomy conjecture has attracted a great deal of attention lately (see for
example [3, 4, 5, 7, 9, 12, 22, 23, 24, 25, 31]), and the conjecture has been settled in
various special cases. A deep connection with universal algebra was uncovered by
P. Jeavons [22] and further refined in collaboration with Bulatov and Krokhin [7, 8].
They have stated a precise conjecture predicting which sets of relations should give
rise to tractable problems and which are NP-complete (a conjecture along the same
lines is sketched in [12].) We now state the precise conjecture and briefly outline
the necessary background (we refer the reader to [20], [32] and [34] for basic results
in universal algebra.)
Let f be an n-ary operation on a set A and let θ be a k-ary relation on A. We
say that f preserves θ or that θ is invariant under f if the following holds: given
any matrix M of size k × n whose columns are in θ, applying f to the rows of M
will produce a k-tuple in θ. Given a set R of relations on A, we define P ol(R) to
be the set of all operations on A that preserve all relations in R. Jeavons observed
that the complexity of the problem CSP (R) is essentially determined by P ol(R).
Bulatov, Jeavons and Krokhin prove that, to settle the dichotomy conjecture, it
is sufficient to consider sets of relations R such that every member of P ol(R) is
surjective [7]. When a set of relations has this property we say that it is a core. 1
An operation f on A of arity at least 2 is idempotent if it satisfies the identity
f(x, . . . , x) ≈ x (where ≈ indicates the equality holds with all variables universally
quantified). Equivalently, an operation on A is idempotent if it preserves every
1 More precisely, we should say that the relational structure 〈A; R〉 is a core, i.e. that all its
endomorphisms are automorphisms, see for instance [12] or [19]. Every finite relational structure
has, up to isomorphism, a unique core, and the CSP problems for the structure and its core are
poly-time equivalent [7].