The Complexity of Homomorphism Factorization Kevin M. Berg - - PowerPoint PPT Presentation

The Complexity of Homomorphism Factorization

Kevin M. Berg

University of Colorado Boulder

August 7, 2018

Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 1 / 35

The Homomorphism Factorization Problem

We assume throughout that all algebras are finite. Fix an algebraic language L.

Problem (The Homomorphism Factorization Problem)

Given a homomorphism f : X → Z between L-algebras X and Z and an intermediate L-algebra Y , decide whether there are homomorphisms g : X → Y and h: Y → Z such that f = hg. X Z Y

∃g? f ∃h?

Figure: The general form of the commutative diagram for Homomorphism Factorization Problems.

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Variants on the Homomorphism Factorization Problem

Problem (I. The Homomorphism Problem)

When |Z| = 1, the homomorphisms f and h from the HFP must be constant, reduces to the problem of deciding whether, given L-algebras X and Y , there is a homomorphism g : X → Y . X

• Y

∃g? f (x)= • h(x)= •

Figure: The commutative diagram for the Homomorphism Problem.

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Variants on the Homomorphism Factorization Problem

Problem (II. The Exists Right-Factor Problem)

Given L-algebras X, Y , and Z, and homomorphisms f : X → Z and h: Y → Z, decide whether there is a homomorphism g : X → Y such that f = hg. X Z Y

∃g? f h

Figure: The commutative diagram for the Exists Right-Factor Problem.

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Variants on the Homomorphism Factorization Problem

Problem (II. The Exists Right-Factor Problem)

Given L-algebras X, Y , and Z, and homomorphisms f : X → Z and h: Y → Z, decide whether there is a homomorphism g : X → Y such that f = hg. X Z Y

∃g? f h

Figure: The commutative diagram for the Exists Right-Factor Problem.

Note that the Homomorphism Problem is a special case of the Exists Right-Factor Problem.

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Variants on the Homomorphism Factorization Problem

Problem (III. The Exists Left-Factor Problem)

Given L-algebras X, Y , and Z, and homomorphisms f : X → Z and g : X → Y , decide whether there is a homomorphism h: Y → Z such that f = hg. X Z Y

g f ∃h?

Figure: The commutative diagram for the Exists Left-Factor Problem.

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Variants on the Homomorphism Factorization Problem

Problem (IV. The Retraction Problem)

When Z = X, and f is the identity function, reduces to the problem of deciding if, given X and Y , the algebra X is a retract of Y . X X Y

∃g? id ∃h?

Figure: The commutative diagram for the Retraction Problem.

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Variants on the Homomorphism Factorization Problem

Problem (V. The Isomorphism Problem)

Restrict the retraction problem to the special case where |X| = |Y |. X X Y

∃g? id ∃h?

Figure: The commutative diagram for the Isomorphism Problem.

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Original MathOverflow Question

The following question was posted to MathOverflow in February 2017:

Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 8 / 35

Original MathOverflow Question

The following question was posted to MathOverflow in February 2017:

Problem (Van Name, ’17)

Let X, Y , and Z, be finite algebras with a single binary operation. Suppose f : X → Z and h: Y → Z are homomorphisms. Is there an

• ptimized computer program that searches for homomorphisms g : X → Y

where f = hg? Is the problem of finding such a homomorphism g NP-Complete? Is this problem still NP-Complete when all operations are associative?

Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 8 / 35

Original MathOverflow Question

The following question was posted to MathOverflow in February 2017:

Problem (Van Name, ’17)

Let X, Y , and Z, be finite algebras with a single binary operation. Suppose f : X → Z and h: Y → Z are homomorphisms. Is there an

• ptimized computer program that searches for homomorphisms g : X → Y

where f = hg? Is the problem of finding such a homomorphism g NP-Complete? Is this problem still NP-Complete when all operations are associative? We will show that determining whether such a g exists is NP-Complete, even for algebras with associative binary operations.

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Computational Complexity of HFPs

Increasing Generality Homomorphism Factorization

Computational Complexity of HFPs

Increasing Generality Homomorphism Factorization Exists Right-Factor Exists Left-Factor Retraction Problem

Computational Complexity of HFPs

Increasing Generality Homomorphism Factorization Exists Right-Factor Exists Left-Factor Retraction Problem Homomorphism Problem Isomorphism Problem

Computational Complexity of HFPs

Increasing Generality Homomorphism Factorization Exists Right-Factor Exists Left-Factor Retraction Problem Homomorphism Problem Isomorphism Problem

B: Non-Associative Binary

NP NP GI GI B B B B B B B B

Computational Complexity of HFPs

Increasing Generality Homomorphism Factorization Exists Right-Factor Exists Left-Factor Retraction Problem Homomorphism Problem Isomorphism Problem

B: Non-Associative Binary

NP NP GI GI B B B B B B B B

S: Semigroups

P S S S S S S Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 9 / 35

Graph Homomorphism

Definition (Undirected Graph, G)

G = (VG, EG) is a relational structure consisting of a universe, VG, of vertices, together with a symmetric binary relation, EG, the set of edges of G. Unless stated otherwise, we assume all graphs are loopless.

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Graph Homomorphism

Definition (Undirected Graph, G)

G = (VG, EG) is a relational structure consisting of a universe, VG, of vertices, together with a symmetric binary relation, EG, the set of edges of G. Unless stated otherwise, we assume all graphs are loopless.

Theorem (Graph Homomorphism)

Given two finite graphs, G and H, the question of whether there exists a relational homomorphism φ: G → H is NP-Complete.

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Graph Homomorphism

Definition (Undirected Graph, G)

G = (VG, EG) is a relational structure consisting of a universe, VG, of vertices, together with a symmetric binary relation, EG, the set of edges of G. Unless stated otherwise, we assume all graphs are loopless.

Theorem (Graph Homomorphism)

Given two finite graphs, G and H, the question of whether there exists a relational homomorphism φ: G → H is NP-Complete.

Theorem (Strong Graph Homomorphism)

Given two finite graphs, G and H, the question of whether there exists a strong relational homomorphism φ: G → H is NP-Complete.

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Non-Associative Case

Let G = (VG, EG) be an undirected graph.

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Non-Associative Case

Let G = (VG, EG) be an undirected graph.

Definition (G ∗)

For every v in VG, there are two elements, v1 and v2 in G ∗. There are also four distinguished elements, a, b, c, and d. We then assign to G ∗ a non-associative binary operation, ·, to be defined.

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Encoding Example: C ∗

4

w1 x1 y1 z1

Encoding Example: C ∗

4

w1 x1 y1 z1 a

Encoding Example: C ∗

4

w1 x1 y1 z1 a w2 x2 y2 z2

Encoding Example: C ∗

4

w1 x1 y1 z1 a w2 x2 y2 z2 b

Encoding Example: C ∗

4

w1 x1 y1 z1 a w2 x2 y2 z2 b

Encoding Example: C ∗

4

w1 x1 y1 z1 a w2 x2 y2 z2 b c

Encoding Example: C ∗

4

w1 x1 y1 z1 a w2 x2 y2 z2 b c d

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Multiplication Table for G ∗

For any distinct u, v in VG, we have · a b c d u1 v1 u2 v2 a b a a a u1 v1 u2 v2 b a c a a u1 v1 u2 v2 c a a d a u1 v1 u2 v2 d a a a a u1 v1 u2 v2 u1 u1 u1 u1 u1 d ∗ c d v1 v1 v1 v1 v1 ∗ d d c u2 u2 u2 u2 u2 c d d b v2 v2 v2 v2 v2 d c b d where ∗ is either u1v1 = v1u1 = a if (u, v) is in EG, or else u1v1 = v1u1 = d.

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Finite Algebras with a Non-Associative Binary Operation

Theorem (B., ’18)

Let G and H be undirected graphs with at least two vertices. There exists a homomorphism ψ: G ∗ → H∗ if and only if there exists a strong graph homomorphism φ: G → H.

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Finite Algebras with a Non-Associative Binary Operation

Theorem (B., ’18)

Let G and H be undirected graphs with at least two vertices. There exists a homomorphism ψ: G ∗ → H∗ if and only if there exists a strong graph homomorphism φ: G → H.

Corollary

The Homomorphism Problem for finite algebras with a single binary

• peration is NP-Complete.

Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 14 / 35

Finite Algebras with a Non-Associative Binary Operation

Theorem (B., ’18)

Let G and H be undirected graphs with at least two vertices. There exists a homomorphism ψ: G ∗ → H∗ if and only if there exists a strong graph homomorphism φ: G → H.

Corollary

The Homomorphism Problem for finite algebras with a single binary

• peration is NP-Complete.

Corollary

The Exists Right-Factor Problem for finite algebras with a single binary

• peration is NP-Complete.

Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 14 / 35

Finite Algebras with a Non-Associative Binary Operation

Theorem (B., ’18)

Let G and H be undirected graphs with at least two vertices. There exists a homomorphism ψ: G ∗ → H∗ if and only if there exists a strong graph homomorphism φ: G → H.

Corollary

The Homomorphism Problem for finite algebras with a single binary

• peration is NP-Complete.

Corollary

The Exists Right-Factor Problem for finite algebras with a single binary

• peration is NP-Complete.

Theorem

The Retraction Problem and Exists Left-Factor Problems for finite algebras with a single binary operation is NP-Complete.

Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 14 / 35

Complications

This approach, wherein we reduce the Homomorphism Problem to the problem of Graph Homomorphism, does not work for semigroups.

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Complications

This approach, wherein we reduce the Homomorphism Problem to the problem of Graph Homomorphism, does not work for semigroups.

Proposition

Every finite semigroup has an idempotent.

Corollary

The Homomorphism Problem for finite semigroups is in polynomial time.

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Complications

This approach, wherein we reduce the Homomorphism Problem to the problem of Graph Homomorphism, does not work for semigroups.

Proposition

Every finite semigroup has an idempotent.

Corollary

The Homomorphism Problem for finite semigroups is in polynomial time. Therefore, to show that the Exists Right-Factor Problem is NP-Complete for semigroups, we must work with the factorization.

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Graph Encoding into Semigroups

Let G = (VG, EG) be an undirected graph.

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Graph Encoding into Semigroups

Let G = (VG, EG) be an undirected graph.

Definition (Semigroup XG)

The universe of XG consists of an element, v, for each v in VG; an element, χu,v, for each u, v in VG such that (u, v) is not an element of EG (let χu,v = χv,u); and auxiliary elements b, b2, c, and 0. We assign to XG the single binary operation, ·, to be defined.

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Encoding Example: XC4

w x y z

Encoding Example: XC4

w x y z b

Encoding Example: XC4

w x y z b

Encoding Example: XC4

w x y z b c

Encoding Example: XC4

w x y z b c b2

Encoding Example: XC4

w x y z b c b2 χw,w, χw,y, . . . , 0

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Multiplication Table for XG

For any distinct u, v in VG, we have · b b2 c u v χ b b2 c c b2 c u c χu,u ∗ v c ∗ χv,v χ where for any u and v in VG, ∗ is either uv = vu = c if (u, v) is in EG, or else uv = vu = χu,v; and χ is a placeholder for any χu,v in the semigroup.

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The Special Semigroup Z

We define a target semigroup Z by encoding the graph consisting of a single loop on a vertex a.

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The Special Semigroup Z

We define a target semigroup Z by encoding the graph consisting of a single loop on a vertex a.

Definition (Z Multiplication Table)

· a b b2 c a c c b c b2 b2 c

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Encoding Z

a b c b2

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Homomorphism Diagram

G b ·0 X H b ·0 Y a b ·0 Z f h

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Homomorphism Diagram

G b ·0 X H b ·0 Y a b ·0 Z f (b2) = b2

Homomorphism Diagram

G b ·0 X H b ·0 Y a b ·0 Z f (b2) = b2 f (b) = b

Homomorphism Diagram

G b ·0 X H b ·0 Y a b ·0 Z f (b2) = b2 f (b) = b f (u) = f (v) = a

Homomorphism Diagram

G b ·0 X H b ·0 Y a b ·0 Z f (b2) = b2 f (b) = b f (u) = f (v) = a f (c) = f (χ) = c

Homomorphism Diagram

G b ·0 X H b ·0 Y a b ·0 Z f (b2) = b2 f (b) = b f (u) = f (v) = a f (c) = f (χ) = c f (0) = 0

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Homomorphism Diagram

G b ·0 X H b ·0 Y a b ·0 Z f h

Homomorphism Diagram

G b ·0 X H b ·0 Y a b ·0 Z f h ∃g?

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NP-Complete

Theorem (B., ’18)

There exists a semigroup homomorphism g : X → Y with f = hg if and

• nly if there exists a graph homomorphism φ: G → H.

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NP-Complete

Theorem (B., ’18)

There exists a semigroup homomorphism g : X → Y with f = hg if and

• nly if there exists a graph homomorphism φ: G → H.

Corollary

The Exists Right-Factor Problem for finite semigroups is NP-Complete. With some modification, this construction can also be used to show:

Theorem

The Exists Left-Factor Problem for finite semigroups is NP-Complete.

Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 24 / 35

NP-Complete

Theorem (B., ’18)

There exists a semigroup homomorphism g : X → Y with f = hg if and

• nly if there exists a graph homomorphism φ: G → H.

Corollary

The Exists Right-Factor Problem for finite semigroups is NP-Complete.

Corollary

The Retraction Problem for finite semigroups is NP-Complete. With some modification, this construction can also be used to show:

Theorem

The Exists Left-Factor Problem for finite semigroups is NP-Complete.

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A Strategy for Polynomial Time Cases

Recall the general commutative diagram for Homomorphism Factorization Problems: X Z Y

∃g? f ∃h?

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A Strategy for Polynomial Time Cases

Suppose that there exists a retraction r : X → X with the following property:

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A Strategy for Polynomial Time Cases

Suppose that there exists a retraction r : X → X with the following property:

Proposition (Respects f )

A retraction r : X → X respects a homomorphism f : X → Z if fr = f .

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A Strategy for Polynomial Time Cases

Suppose that there exists a retraction r : X → X with the following property:

Proposition (Respects f )

A retraction r : X → X respects a homomorphism f : X → Z if fr = f . X Z Y

r g f h

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A Strategy for Polynomial Time Cases

Let X ′ = r(X). We have:

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A Strategy for Polynomial Time Cases

Let X ′ = r(X). We have: X X ′ Z Y

g f ⊇ r g′ f |X′=f h

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A Strategy for Polynomial Time Cases

Let X ′ = r(X). We have: X X ′ Z Y

g f ⊇ r g′ f |X′=f h

If r respects f , then fr = f and therefore f |X ′ = f . Consequently,

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A Strategy for Polynomial Time Cases

Let X ′ = r(X). We have: X X ′ Z Y

g f ⊇ r g′ f |X′=f h

If r respects f , then fr = f and therefore f |X ′ = f . Consequently,

Proposition

f factors through Y if and only if f |X ′ factors through Y .

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A Strategy for Polynomial Time Cases

Definition (f -Core)

A is an f -core of X if A is minimal with respect to the existence of an

• nto, f -respecting retraction, r : X → A. If X is its own f -core, X is an

f -core.

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A Strategy for Polynomial Time Cases

Definition (f -Core)

A is an f -core of X if A is minimal with respect to the existence of an

• nto, f -respecting retraction, r : X → A. If X is its own f -core, X is an

f -core.

Definition (Bounded f -Core)

Variety V has bounded f -cores if, for any finite algebra X in V and any surjective map f : X → Z for which X is an f -core, |X| ≤ m(|Z|) for some function m.

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Bounded f -Cores and the Exists Right-Factor Problem

When the target algebra Z is fixed, with bounded f -cores:

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Bounded f -Cores and the Exists Right-Factor Problem

When the target algebra Z is fixed, with bounded f -cores: X f Z ∃g? h Y There are |Y ||X ′| ≤ |Y |m(|Z|) many choices for g; polynomial in |Y |.

Bounded f -Cores and the Exists Right-Factor Problem

When the target algebra Z is fixed, with bounded f -cores: X f Z ∃g? h Y X ′

Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 29 / 35

Bounded f -Cores and the Exists Right-Factor Problem

When the target algebra Z is fixed, with bounded f -cores: X f Z ∃g? h Y X ′ There are |Y ||X ′| ≤ |Y |m(|Z|) many choices for g; polynomial in |Y |.

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Bounded f -Cores and Polynomial Time Cases

Theorem

The Exists Right-Factor problem can be solved in polynomial time for a variety V if the following conditions hold:

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Bounded f -Cores and Polynomial Time Cases

Theorem

The Exists Right-Factor problem can be solved in polynomial time for a variety V if the following conditions hold:

I.

V has bounded f -cores.

Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 30 / 35

Bounded f -Cores and Polynomial Time Cases

Theorem

The Exists Right-Factor problem can be solved in polynomial time for a variety V if the following conditions hold:

I.

V has bounded f -cores.

II.

The f -cores of finite algebras in V can be found in polynomial time.

Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 30 / 35

Bounded f -Cores and Polynomial Time Cases

Theorem

The Exists Right-Factor problem can be solved in polynomial time for a variety V if the following conditions hold:

I.

V has bounded f -cores.

II.

The f -cores of finite algebras in V can be found in polynomial time.

III.

Given a finite algebra in V, a retraction from X to its f -core can be found in polynomial time.

Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 30 / 35

Condition I: Varieties with Bounded f -Cores

Let f be a function appropriately defined for a given variety.

Proposition (G-Sets)

Let G be a finite group. Then the variety of G-sets has bounded f -cores.

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Condition I: Varieties with Bounded f -Cores

Let f be a function appropriately defined for a given variety.

Proposition (G-Sets)

Let G be a finite group. Then the variety of G-sets has bounded f -cores.

Proposition (Boolean Algebras)

Boolean algebras have bounded f -cores.

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Condition I: Varieties with Bounded f -Cores

Let f be a function appropriately defined for a given variety.

Proposition (G-Sets)

Let G be a finite group. Then the variety of G-sets has bounded f -cores.

Proposition (Boolean Algebras)

Boolean algebras have bounded f -cores.

Proposition (Vector Spaces)

Let F be a field. Then the variety of vector spaces over F has bounded f -cores.

Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 31 / 35

Condition I: Varieties with Bounded f -Cores

Let f be a function appropriately defined for a given variety.

Proposition (G-Sets)

Let G be a finite group. Then the variety of G-sets has bounded f -cores.

Proposition (Boolean Algebras)

Boolean algebras have bounded f -cores.

Proposition (Vector Spaces)

Let F be a field. Then the variety of vector spaces over F has bounded f -cores.

Proposition (Abelian Groups)

The variety of abelian groups has bounded f -cores.

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A Variety without Bounded f -Cores

The variety of semilattices does not have bounded f -cores.

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A Variety without Bounded f -Cores

The variety of semilattices does not have bounded f -cores.

Theorem (B., ’18)

Consider the semilattice Z = ({A, B, C, 0}, ∧) given by A ∧ B = B ∧ C = A ∧ C = 0. Then for any natural number n, there exists a semilattice, X, of size at least n that is an f -core for a surjective f : X → Z.

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An Unbounded Semilattice f -Core

A B C Z

An Unbounded Semilattice f -Core

A B C Z b a1 a2 . . . an va1 vc2 va2 . . . vcn van c1 c2 . . . cn f X

An Unbounded Semilattice f -Core

A B C Z b a1 a2 . . . an va1 vc2 va2 . . . vcn van c1 c2 . . . cn f X

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Further Complications

Theorem

The problem of finding the f -core of an arbitrary semigroup is NP-Complete.

Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 34 / 35

Further Complications

Theorem

The problem of finding the f -core of an arbitrary semigroup is NP-Complete.

Conjecture (A Possibility)

The Exists Right-Factor Problem is in polynomial time for a given variety if and only if the variety has bounded f -cores.

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Some Open Questions

Problem (Open)

Can the retraction map of X onto its f -core be determined in polynomial time if the f -core is already known?

Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 35 / 35

Some Open Questions

Problem (Open)

Can the retraction map of X onto its f -core be determined in polynomial time if the f -core is already known?

Problem (Open)

What conditions must a variety satisfy to always have bounded f -cores? For those that do not, what conditions (if any) might be required of Z?

Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 35 / 35

Some Open Questions

Problem (Open)

Can the retraction map of X onto its f -core be determined in polynomial time if the f -core is already known?

Problem (Open)

What conditions must a variety satisfy to always have bounded f -cores? For those that do not, what conditions (if any) might be required of Z?

Problem (Open)

Are the Exists Right-Factor and Exists Left-Factor Problems NP-Complete for algebras with only unary operations?

Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 35 / 35

Some Open Questions

Problem (Open)

Can the retraction map of X onto its f -core be determined in polynomial time if the f -core is already known?

Problem (Open)

What conditions must a variety satisfy to always have bounded f -cores? For those that do not, what conditions (if any) might be required of Z?

Problem (Open)

Are the Exists Right-Factor and Exists Left-Factor Problems NP-Complete for algebras with only unary operations?

Problem (Open)

Is the Retraction Problem NP-Complete for groups?

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