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# Csr2011 june16 11_00_carvalho

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### Csr2011 june16 11_00_carvalho

1. 1. Maltsev digraphsC. Carvalhoa , L. Egrib , M. Jacksonc and T. Nivenca University of Hertfordshire, b McGill University, c LaTrobe University CSR 2011
2. 2. DigraphsA directed graph or digraph is a relational structure G = (V , E)with domain V , the vertices, and relation E ⊆ V × V , thedirected edges or arcs.A polymorphism f of G is an n-ary operation in V that is ahomomorphism f : G × · · · × G −→ G.A maltsev polymorphism of G is a 3-ary polymorphism, m, thatsatisﬁes m(x, y , y ) = m(y , y , x) = x for all x, y ∈ V .
3. 3. Constraint Satisfaction Problem or Homomorphism ProblemGiven two digraphs G = (V (G), E(G)), H = (V (H), E(H)) isthere a homomorphism h : G −→ H?h satisﬁes (a, b) ∈ E(G) ⇒ (h(a), h(b)) ∈ E(H).If a homomorphism exists we write G −→ H.The CSP with a ﬁxed H is denoted CSP(H), and can be viewedas {G : G −→ H}.
4. 4. CSPExampleCSP(K2 ) is the graph 2-colourability problem.More generally CSP(H), with H a digraph is the H-colouringproblem. This is a much studied problem in graph theory.Problem: Classify CSP(H) with respect to computational (anddescriptive) complexity.
5. 5. Polymorphisms and complexityGood polymorphisms imply tractability.Lack of polymorphisms NP-complete problems.Certain polymorphism imply membership in a speciﬁccomplexity class.Example (Dalmau, Krokhin)If H is preserved by a majority operation then CSP(H) is in NL.m(x, y , y ) = m(y , x, y ) = m(y , y , x) = y
6. 6. Polymorphisms and complexityGood polymorphisms imply tractability.Lack of polymorphisms NP-complete problems.Certain polymorphism imply membership in a speciﬁccomplexity class.Example (Dalmau, Krokhin)If H is preserved by a majority operation then CSP(H) is in NL.m(x, y , y ) = m(y , x, y ) = m(y , y , x) = y
7. 7. Example (Dalmau, Larose)If H is preserved by a maltsev operation + extra condition thenCSP(H) is in L.Polymorphisms are not easy to understand from acombinatorial point of view.Aim: combinatorial description of digraphs with certainalgebraic properties.
8. 8. CharacterizationTheoremA digraph is preserved by a maltsev polymorphism iff it doesnot contain an N of any length s s Ts d T s s s ds s T T s d T d T s s ds s dsExampleThe graph •0 / •1 has no maltsev since it contains •O0 • ? O1   •0 •1
9. 9. CharacterizationTheoremA digraph is preserved by a maltsev polymorphism iff it doesnot contain an N of any length s s Ts d T s s s ds s T T s d T d T s s ds s dsExampleThe graph •0 / •1 has no maltsev since it contains •O0 • ? O1   •0 •1
10. 10. Recognition in polynomial timeThe non existence of Ns of length 1 is equivalent in theadjacency matrix to the following property: when 2 rows, orcolumns, share a 1 then they are identical.It folllows from that maltsev digraphs can be recognized inpolynomial time on the number of vertices, O(n4 ).
11. 11. CSPTheoremLet H be a maltsev digraph. If H is acyclic then it retracts onto adirected path. Otherwise it retracts onto the disjoint union ofdirected cycles.It then follows that, for a maltsev digraph H, CSP(H) is in L.
12. 12. List Homomorphism problemThe list H-colouring problem, given an input graph G and foreach vertex v ∈ V (G) a list Lv ⊆ V (H), asks if there isf : G −→ H s.t. f (v ) ∈ Lv .It is equivalent to CSP(Hu ), where Hu is H together with allunary relations {S : S ⊆ V (H)}.In this case we need conservative polymorphisms, i.e.m(x, y , z) ∈ {x, y , z}.
13. 13. CharacterizationTheoremA digraph admits a conservative maltsev polymorphism iff  x → · · · → xk −1 → u  ∃ x → · · · → wk −1 → v with y → · · · → yk −1 → u ⇒ wi ∈ {xi , yi , zi } for each i. y → · · · → zk −1 → v 
14. 14. ExampleThe digraph s s T s d  T s ds   s s  rB ¨ T¨¨rr T s ¨ rshas a maltsev polymorphism, but not a conservative maltsevpolymorphism.We can then deduce that, when H is a maltsev digraph,LHOM(H) is in L.
15. 15. Cycles and pathsTheoremLet H be a (conservative) maltsev digraph. Then H ispreserved by (conservative) polymorphisms t1 , t2 , . . . , tk (notnecessarily distinct) satisfying a single equational sequence t1 (x1,1 , x1,2 , . . . , x1,n1 ) ≈ · · · ≈ tk (xk ,1 , xn,2 , . . . , xk ,nk ) ≈ x,where {x1,1 , . . . , x1,n1 } = · · · = {xk ,1 , . . . , xk ,nk } andx ∈ {x1,1 , . . . , x1,n1 }, iff the directed path/cycle is preserved bythese polymorphisms.Maltsev iff Pixley iff Minority implies Majoritym(x, y , y ) = m(y , y , x) = m(x, y , x) = xm(x, y , y ) = m(y , x, y ) = m(y , y , x) = x
16. 16. 2-semilattice polymorphismsA binary polymorphism c is commutative if c(x, y ) = c(y , x).A disjoint union of directed cycles admits a binary conservativecommutative polymorphism iff it contains no even cycles.TheoremA maltsev digraph with a conservative commutativepolymorphism is a conservative maltsev digraph.These do not retract to cycles of even length.