Caterpillar duality for CSP. C. Carvalho (UoD), V. Dalmau (UPF) and A. Krokhin (UoD)

Transcription

1 Caterpillar duality for CSP C. Carvalho (UoD), V. Dalmau (UPF) and A. Krokhin (UoD)

2 Motivation The following problems Scheduling; System of linear equations; Drawing up a timetable; Check the satisfiability of a logical formula; Finding Hamiltonian cycles; Choosing frequencies for a mobile-phone network; Graph colourability; involve searching for a solution which satisfies certain constraints. They are all examples of Constraint Satisfaction Problems, CSP.

3 The Constraint Satisfaction Problem The Homomorphism Problem Given two finite relational structures A = (A; R 1 A,..., R m A ) and B = (B; R 1 B,..., R m B ) is there a homomorphism h : A B? The CSP with a a fixed B is denoted CSP(B), and can be viewed as {A : A B}. Problems: Classify CSP(B) with respect to computational complexity; Classify CSP(B) with respect to descriptive complexity; Determine complexity of deciding if CSP(B) has a given complexity (meta-problem).

4 Obstruction sets An obstruction set for a structure B is a class O B of structures such that, for all structures A Example A B iff A A for all A O B. If B is a bipartite graph then O B can be chosen to consist of all odd cycles. Obs B A B

5 Dualities A structure B has nice duality if O B can be chosen to be simple: Nice= Finite, Simple= finite Ex: B is a transitive tournament; Nice= Path, Simple= consisting of "paths" Ex: B is an oriented path (Hell, Zhu 94); Nice= Caterpillar, Simple= consisting of... Ex:...later... Nice= Tree, Simple= consisting of "trees" Ex: B is an oriented tree that is preserved by a min order (Hell, Nesestril, Zhu 96). More info on Dualities for constraint satisfaction problems, survey by Bulatov, Krokhin, Larose.

6 Caterpillars In graph theory a caterpillar is a tree where all vertices are within distance 1 of a central path. The notions of tree, caterpillar and path can be extended to arbitrary structures, by means of the incidence multigraph of a structure. Roughly, the vertices represent elements and tuples of the relations and the edges represent containment.

7 Example The structure A with domain {1,..., 6} and relations R 1 = {2, 3}, R 2 = {(1, 2), (2, 3), (3, 6)}, R 3 = {(3, 4, 5)} can be decomposed as a caterpillar. (R 1,2) (R 1,3) 5 1 (R 2,(1,2)) 2 (R 2,(2,3)) 3 4 (R 3,(3,4,5)) (R 2,(3,6)) 6

8 Datalog For logical characterizations we use Datalog. Datalog program for co-csp(k 2 ) oddpath(x, y) : edge(x, y) oddpath(x, y) : oddpath(x, z), edge(z, w), edge(w, y) oddcycle : oddpath(x, x) EDB s are relations from the structure -edge(x, y), IDB s are auxiliary predicates -oddpath(x, y), linear Datalog has at most one IDB in the body of each rule, monadic Datalog has at most unary IDB s, co-csp(b) is definable by a Datalog program iff the program accepts only structures A s.t. A B. If co-csp(b) is definable in (linear) Datalog then CSP(B) is in (NL) PTIME.

9 Polymorphisms An n-ary operation f : B n B is a polymorphism of a relation R if for any tuples a 1,..., a n R the tuple obtained by applying f coordinate-wise is also in R; f is a polymorphism of the structure B if it is a polymorphism of all relations in B. Example The binary operation min is a polymorphism of R = {(0, 0, 0), (1, 0, 0), (0, 0, 1)} min((1, 0, 0), (0, 0, 1)) = min((1, 0, 0), (0, 0, 1)) = (min(1, 0), min(0, 0), min(0, 1)) = (0, 0, 0) R

10 Some useful operations An operation f is idempotent if f (x,..., x) = x, conservative if f (x 1,..., x n ) {x 1,..., x n }, totally symmetric if f (x 1,..., x n ) = f (y 1,..., y n ) whenever {x 1,..., x n } = {y 1,..., y n }, k-block symmetric if f (S 1,..., S n ) = f (T 1,..., T n ), whenever {S 1,..., S n } = {T 1,..., T n }, with S i = {x i1,..., x ik }, k-abs operation if it is k-block symmetric and it satisfies the absorptive rule f (S 1, S 2, S 3,..., S n ) = f (S 2, S 2, S 3,..., S n ) whenever S 2 S 1.

11 Examples Example Oriented paths have polymorphisms min(x 1,..., x n ) for every n Example Oriented paths have k-abs polymorphisms min(max(x 11,..., x 1k ),..., max(x n1,..., x nk )), for all k, n 1.

12 Tree duality The following result characterizes structures with tree duality Theorem (Feder, Vardi / Dalmau, Pearson) Tfae 1. B has tree duality; 2. co-csp(b) is definable by a monadic Datalog program with at most one EDB per rule; 3. U(B) B; 4. B has n-ary totally symmetric polymorphism, for all n 1; 5. if B is a core then it is the core of a structure with a semilattice polymorphism.

13 Theorem Tfae 1. B has caterpillar duality; Caterpillar duality 2. co-csp(b) is definable by a linear monadic Datalog program with at most one EDB per rule; 3. C(B) B; 4. B has (kn)-ary k-abs polymorphism, for all n, k 1; 5. if B is a core then it is the core of a structure with lattice polymorphisms. It gives a logical, a combinatorial and an algebraic characterization of structures with caterpillar duality. It also answers the meta-problem for this duality: it is decidable but NP-hard.

14 Caterpillar vs path duality Theorem There exists a structure with caterpillar duality, but not path duality. The fragment of Datalog arising from path duality is not as natural. A natural algebraic characterization of path duality is unlikely. Caterpillar duality seems a more robust notion.

15 Applications to digraph H-colouring H-colouring is equivalent to CSP(H) The min and max polymorphisms of oriented paths can be rearranged and extended to yield: Theorem If H is a planar layered directed acyclic digraph then H has k-abs polymorphisms of arity kn for all k, n 1. Consequently H has caterpillar duality. Example Oriented caterpillars have caterpillar duality.

16 Applications to list H-colouring The list H-colouring problem is equivalent to CSP(H u ), where H u is H together with all unary relations {S : S H}. An interval graph is a reflexive graph whose vertices can be represented by intervals in R, s.t. vertices are adjacent iff the intervals intersect. Theorem Interval graphs have conservative k-abs polymorphisms of arity kn, for every k, n 1. As a consequence of a result by Feder and Hell, we have Corollary For a reflexive graph H, either H u has caterpillar duality or CSP(H u ) is NP-complete.