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 . f X Z ∃ g ? ∃ h ? Y Figure: The general form of the commutative diagram for Homomorphism Factorization Problems. Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 2 / 35
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 . f ( x )= • • X ∃ g ? h ( x )= • Y Figure: The commutative diagram for the Homomorphism Problem. Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 3 / 35
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 . f X Z ∃ g ? h Y Figure: The commutative diagram for the Exists Right-Factor Problem. Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 4 / 35
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 . f X Z ∃ g ? h Y 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. Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 4 / 35
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 . f X Z g ∃ h ? Y Figure: The commutative diagram for the Exists Left-Factor Problem. Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 5 / 35
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 . id X X ∃ g ? ∃ h ? Y Figure: The commutative diagram for the Retraction Problem. Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 6 / 35
Variants on the Homomorphism Factorization Problem Problem (V. The Isomorphism Problem) Restrict the retraction problem to the special case where | X | = | Y | . id X X ∃ g ? ∃ h ? Y Figure: The commutative diagram for the Isomorphism Problem. Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 7 / 35
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 optimized 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 optimized 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. Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 8 / 35
Computational Complexity of HFPs Homomorphism Factorization Increasing Generality
Computational Complexity of HFPs Homomorphism Factorization Increasing Exists Exists Retraction Generality Right-Factor Left-Factor Problem
Computational Complexity of HFPs Homomorphism Factorization Increasing Exists Exists Retraction Generality Right-Factor Left-Factor Problem Homomorphism Isomorphism Problem Problem
Computational Complexity of HFPs Homomorphism Factorization B: Non-Associative Binary B NP NP Increasing Exists Exists Retraction Generality Right-Factor Left-Factor Problem GI GI B B B B Homomorphism Isomorphism Problem Problem B B B
Computational Complexity of HFPs Homomorphism Factorization B: Non-Associative Binary S: Semigroups B S NP NP Increasing Exists Exists Retraction Generality Right-Factor Left-Factor Problem GI GI B S B S B B S P Homomorphism Isomorphism Problem Problem B S B B S Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 9 / 35
Graph Homomorphism Definition (Undirected Graph, G ) G = ( V G , E G ) is a relational structure consisting of a universe, V G , of vertices, together with a symmetric binary relation, E G , the set of edges of G . Unless stated otherwise, we assume all graphs are loopless. Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 10 / 35
Graph Homomorphism Definition (Undirected Graph, G ) G = ( V G , E G ) is a relational structure consisting of a universe, V G , of vertices, together with a symmetric binary relation, E G , 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. Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 10 / 35
Graph Homomorphism Definition (Undirected Graph, G ) G = ( V G , E G ) is a relational structure consisting of a universe, V G , of vertices, together with a symmetric binary relation, E G , 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. Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 10 / 35
Non-Associative Case Let G = ( V G , E G ) be an undirected graph. Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 11 / 35
Non-Associative Case Let G = ( V G , E G ) be an undirected graph. Definition ( G ∗ ) For every v in V G , there are two elements, v 1 and v 2 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. Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 11 / 35
Encoding Example: C ∗ 4 x 1 y 1 w 1 z 1
Encoding Example: C ∗ 4 a x 1 y 1 w 1 z 1
Encoding Example: C ∗ 4 x 2 y 2 a w 2 z 2 x 1 y 1 w 1 z 1
Encoding Example: C ∗ 4 x 2 y 2 a w 2 z 2 b x 1 y 1 w 1 z 1
Encoding Example: C ∗ 4 x 2 y 2 a w 2 z 2 b x 1 y 1 w 1 z 1
Encoding Example: C ∗ 4 x 2 y 2 a w 2 z 2 b c x 1 y 1 w 1 z 1
Encoding Example: C ∗ 4 x 2 y 2 a w 2 z 2 b c x 1 y 1 d w 1 z 1 Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 12 / 35
Multiplication Table for G ∗ For any distinct u , v in V G , we have · a b c d u 1 v 1 u 2 v 2 a b a a a u 1 v 1 u 2 v 2 b a c a a u 1 v 1 u 2 v 2 c a a d a u 1 v 1 u 2 v 2 d a a a a u 1 v 1 u 2 v 2 u 1 u 1 u 1 u 1 u 1 d ∗ c d ∗ v 1 v 1 v 1 v 1 v 1 d d c u 2 u 2 u 2 u 2 u 2 c d d b v 2 v 2 v 2 v 2 v 2 d c b d where ∗ is either u 1 v 1 = v 1 u 1 = a if ( u , v ) is in E G , or else u 1 v 1 = v 1 u 1 = d . Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 13 / 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. Kevin M. Berg (CU Boulder) Homomorphism Factorization August 7, 2018 14 / 35
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