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Dichotomy for conservative digraphs Alexandr Kazda Department of - - PowerPoint PPT Presentation

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Introduction Coloring pairs There is no blue pair Combinatorics on potatoes Dichotomy for conservative digraphs Alexandr Kazda Department of Algebra Charles University, Prague June 9th, 2012 Alexandr Kazda Dichotomy for conservative

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Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Dichotomy for conservative digraphs

Alexandr Kazda

Department of Algebra Charles University, Prague

June 9th, 2012

Alexandr Kazda Dichotomy for conservative digraphs

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Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Where are we going

A finite relational structure A is conservative if it contains all possible unary relations. Denote A the algebra of idempotent polymorphisms of A. We show: If A contains a Taylor operation then A generates a congruence meet semidistributive variety. CSP translation: If CSP(A) is not obviously NP-complete, then local consistency checking solves CSP(A).

Alexandr Kazda Dichotomy for conservative digraphs

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Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Where are we going

A finite relational structure A is conservative if it contains all possible unary relations. Denote A the algebra of idempotent polymorphisms of A. We show: If A contains a Taylor operation then A generates a congruence meet semidistributive variety. CSP translation: If CSP(A) is not obviously NP-complete, then local consistency checking solves CSP(A).

Alexandr Kazda Dichotomy for conservative digraphs

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SLIDE 4

Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Where are we going

A finite relational structure A is conservative if it contains all possible unary relations. Denote A the algebra of idempotent polymorphisms of A. We show: If A contains a Taylor operation then A generates a congruence meet semidistributive variety. CSP translation: If CSP(A) is not obviously NP-complete, then local consistency checking solves CSP(A).

Alexandr Kazda Dichotomy for conservative digraphs

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SLIDE 5

Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Where are we going

A finite relational structure A is conservative if it contains all possible unary relations. Denote A the algebra of idempotent polymorphisms of A. We show: If A contains a Taylor operation then A generates a congruence meet semidistributive variety. CSP translation: If CSP(A) is not obviously NP-complete, then local consistency checking solves CSP(A).

Alexandr Kazda Dichotomy for conservative digraphs

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SLIDE 6

Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Where are we going

A finite relational structure A is conservative if it contains all possible unary relations. Denote A the algebra of idempotent polymorphisms of A. We show: If A contains a Taylor operation then A generates a congruence meet semidistributive variety. CSP translation: If CSP(A) is not obviously NP-complete, then local consistency checking solves CSP(A).

Alexandr Kazda Dichotomy for conservative digraphs

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Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Shoulders of giants

  • A. Bulatov: dichotomy for general conservative CSP
  • L. Barto: proof of dichotomy using absorption
  • P. Hell, A. Rafiey: combinatorial characterization of tractable

conservative digraphs which implies our result

Alexandr Kazda Dichotomy for conservative digraphs

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SLIDE 8

Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Shoulders of giants

  • A. Bulatov: dichotomy for general conservative CSP
  • L. Barto: proof of dichotomy using absorption
  • P. Hell, A. Rafiey: combinatorial characterization of tractable

conservative digraphs which implies our result

Alexandr Kazda Dichotomy for conservative digraphs

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SLIDE 9

Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Shoulders of giants

  • A. Bulatov: dichotomy for general conservative CSP
  • L. Barto: proof of dichotomy using absorption
  • P. Hell, A. Rafiey: combinatorial characterization of tractable

conservative digraphs which implies our result

Alexandr Kazda Dichotomy for conservative digraphs

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SLIDE 10

Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Shoulders of giants

  • A. Bulatov: dichotomy for general conservative CSP
  • L. Barto: proof of dichotomy using absorption
  • P. Hell, A. Rafiey: combinatorial characterization of tractable

conservative digraphs which implies our result

Alexandr Kazda Dichotomy for conservative digraphs

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Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Polymorphisms on pairs

If A is conservative and a, b ∈ A then A contains some polymorphism f such that f is semilattice, majority or minority on a, b . . . . . . otherwise all operations on {a, b} are projections. . . . . . and so A has no Taylor operation.

Alexandr Kazda Dichotomy for conservative digraphs

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Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Polymorphisms on pairs

If A is conservative and a, b ∈ A then A contains some polymorphism f such that f is semilattice, majority or minority on a, b . . . . . . otherwise all operations on {a, b} are projections. . . . . . and so A has no Taylor operation.

Alexandr Kazda Dichotomy for conservative digraphs

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SLIDE 13

Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Polymorphisms on pairs

If A is conservative and a, b ∈ A then A contains some polymorphism f such that f is semilattice, majority or minority on a, b . . . . . . otherwise all operations on {a, b} are projections. . . . . . and so A has no Taylor operation.

Alexandr Kazda Dichotomy for conservative digraphs

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SLIDE 14

Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Polymorphisms on pairs

If A is conservative and a, b ∈ A then A contains some polymorphism f such that f is semilattice, majority or minority on a, b . . . . . . otherwise all operations on {a, b} are projections. . . . . . and so A has no Taylor operation.

Alexandr Kazda Dichotomy for conservative digraphs

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SLIDE 15

Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Colors

We color a pair a, b ∈ A: red if it admits a semilattice, else. . . . . . yellow if it admits the majority operation, else. . . . . . we color the pair blue if it admits a minority.

Alexandr Kazda Dichotomy for conservative digraphs

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Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Colors

We color a pair a, b ∈ A: red if it admits a semilattice, else. . . . . . yellow if it admits the majority operation, else. . . . . . we color the pair blue if it admits a minority.

Alexandr Kazda Dichotomy for conservative digraphs

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SLIDE 17

Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Colors

We color a pair a, b ∈ A: red if it admits a semilattice, else. . . . . . yellow if it admits the majority operation, else. . . . . . we color the pair blue if it admits a minority.

Alexandr Kazda Dichotomy for conservative digraphs

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SLIDE 18

Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Colors

We color a pair a, b ∈ A: red if it admits a semilattice, else. . . . . . yellow if it admits the majority operation, else. . . . . . we color the pair blue if it admits a minority.

Alexandr Kazda Dichotomy for conservative digraphs

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Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Colors

Theorem (Bulatov, shortened) There are polymorphisms f (x, y), g(x, y, z), h(x, y, z) ∈ Pol(A) such that for every two-element subset B ⊂ A: f|B is a semilattice operation whenever B is red, and f|B(x, y) = x otherwise, g|B is a majority operation if B is yellow and g|B(x, y, z) = x if B is blue h|B is a minority operation if B is blue.

Alexandr Kazda Dichotomy for conservative digraphs

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Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Colors

Theorem (Bulatov, shortened) There are polymorphisms f (x, y), g(x, y, z), h(x, y, z) ∈ Pol(A) such that for every two-element subset B ⊂ A: f|B is a semilattice operation whenever B is red, and f|B(x, y) = x otherwise, g|B is a majority operation if B is yellow and g|B(x, y, z) = x if B is blue h|B is a minority operation if B is blue.

Alexandr Kazda Dichotomy for conservative digraphs

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Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Colors

Theorem (Bulatov, shortened) There are polymorphisms f (x, y), g(x, y, z), h(x, y, z) ∈ Pol(A) such that for every two-element subset B ⊂ A: f|B is a semilattice operation whenever B is red, and f|B(x, y) = x otherwise, g|B is a majority operation if B is yellow and g|B(x, y, z) = x if B is blue h|B is a minority operation if B is blue.

Alexandr Kazda Dichotomy for conservative digraphs

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Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Colors

Theorem (Bulatov, shortened) There are polymorphisms f (x, y), g(x, y, z), h(x, y, z) ∈ Pol(A) such that for every two-element subset B ⊂ A: f|B is a semilattice operation whenever B is red, and f|B(x, y) = x otherwise, g|B is a majority operation if B is yellow and g|B(x, y, z) = x if B is blue h|B is a minority operation if B is blue.

Alexandr Kazda Dichotomy for conservative digraphs

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Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Blue is bad

If we had no blue vertices, we could use the previous theorem to define 3ary and 4ary WNUs: u(x, y, z) = g(f (f (x, y), z), f (f (y, z), x), f (f (z, x), y)) v(x, y, z, t) = g(f (f (f (x, y), z), t), f (f (f (y, z), x), t)), f (f (f (z, x), y), t)) Then A generates an SD(∧) variety and CSP(A) is easy (see Barto, Kozik).

Alexandr Kazda Dichotomy for conservative digraphs

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Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Blue is bad

If we had no blue vertices, we could use the previous theorem to define 3ary and 4ary WNUs: u(x, y, z) = g(f (f (x, y), z), f (f (y, z), x), f (f (z, x), y)) v(x, y, z, t) = g(f (f (f (x, y), z), t), f (f (f (y, z), x), t)), f (f (f (z, x), y), t)) Then A generates an SD(∧) variety and CSP(A) is easy (see Barto, Kozik).

Alexandr Kazda Dichotomy for conservative digraphs

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Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Blue is bad

If we had no blue vertices, we could use the previous theorem to define 3ary and 4ary WNUs: u(x, y, z) = g(f (f (x, y), z), f (f (y, z), x), f (f (z, x), y)) v(x, y, z, t) = g(f (f (f (x, y), z), t), f (f (f (y, z), x), t)), f (f (f (z, x), y), t)) Then A generates an SD(∧) variety and CSP(A) is easy (see Barto, Kozik).

Alexandr Kazda Dichotomy for conservative digraphs

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Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

There is no blue pair

Assume {a, b} is a blue pair. We can now pp-define the relation R = {(a, a, b), (a, b, a), (b, a, a), (b, b, b)}. This will lead us to a contradiction. . .

Alexandr Kazda Dichotomy for conservative digraphs

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Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

There is no blue pair

Assume {a, b} is a blue pair. We can now pp-define the relation R = {(a, a, b), (a, b, a), (b, a, a), (b, b, b)}. This will lead us to a contradiction. . .

Alexandr Kazda Dichotomy for conservative digraphs

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Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

There is no blue pair

Assume {a, b} is a blue pair. We can now pp-define the relation R = {(a, a, b), (a, b, a), (b, a, a), (b, b, b)}. This will lead us to a contradiction. . .

Alexandr Kazda Dichotomy for conservative digraphs

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Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Combinatorics on potatoes

Assume A has a blue pair and I is the smallest constraint network for R. Then: Each potato contains two or three vertices. Each potato contains only blue pairs. There is no potato with three vertices. There are no interesting relations left and we win.

Alexandr Kazda Dichotomy for conservative digraphs

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Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Combinatorics on potatoes

Assume A has a blue pair and I is the smallest constraint network for R. Then: Each potato contains two or three vertices. Each potato contains only blue pairs. There is no potato with three vertices. There are no interesting relations left and we win.

Alexandr Kazda Dichotomy for conservative digraphs

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SLIDE 31

Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Combinatorics on potatoes

Assume A has a blue pair and I is the smallest constraint network for R. Then: Each potato contains two or three vertices. Each potato contains only blue pairs. There is no potato with three vertices. There are no interesting relations left and we win.

Alexandr Kazda Dichotomy for conservative digraphs

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SLIDE 32

Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Combinatorics on potatoes

Assume A has a blue pair and I is the smallest constraint network for R. Then: Each potato contains two or three vertices. Each potato contains only blue pairs. There is no potato with three vertices. There are no interesting relations left and we win.

Alexandr Kazda Dichotomy for conservative digraphs

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Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Combinatorics on potatoes

Assume A has a blue pair and I is the smallest constraint network for R. Then: Each potato contains two or three vertices. Each potato contains only blue pairs. There is no potato with three vertices. There are no interesting relations left and we win.

Alexandr Kazda Dichotomy for conservative digraphs

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Introduction Coloring pairs There is no blue pair Combinatorics on potatoes

Thanks for your attention.

Alexandr Kazda Dichotomy for conservative digraphs