Universal Algebra and Computational Complexity Lecture 3

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Fix a finite relational structure B. Consider the following problem associated to B: A problem INPUT: a finite structure A in the same signature as B. QUESTION: Is there a homomorphism h : A → B? This problem is called CSP(B). Obviously CSP(B) ∈ NP for any B. Ross Willard (Waterloo) Algebra and Complexity Třešť, September 2008 31 / 31
Fix a finite relational structure B. Consider the following problem associated to B: A problem INPUT: a finite structure A in the same signature as B. QUESTION: Is there a homomorphism h : A → B? This problem is called CSP(B). Obviously CSP(B) ∈ NP for any B. If K 3 is the triangle graph, then CSP(K 3 ) = 3COL, so is NP-complete in this case. If G is a bipartite graph, then then CSP(G) ∈ P. Ross Willard (Waterloo) Algebra and Complexity Třešť, September 2008 31 / 31
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Universal Algebra and Computational
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Summary of Lecture 2 Recall from Tu
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Encoding finite algebras: size matt
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Encoding finite algebras: size matt
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Size of an algebra Define some para
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Some decision problems involving al
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Categories of answers Suppose D is
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Categories of answers Suppose D is
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An easy problem: Subalgebra Members
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An obvious upper bound for SUB-MEM
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An obvious upper bound for SUB-MEM
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The Complexity of SUB-MEM So SUB-ME
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A variation: 1-SUB-MEM 1-SUB-MEM Th
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A variation: 1-SUB-MEM 1-SUB-MEM Th
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Some tractable problems about algeb
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Clone Membership Problem (CLO) INPU
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Clone Membership Problem (CLO) INPU
Page 53 and 54: The Primal Algebra Problem (PRIMAL)
Page 55 and 56: PRIMAL But testing primality of alg
Page 57 and 58: MALTSEV INPUT: a finite algebra A.
Page 59 and 60: Similar characterizations give EXPT
Page 61 and 62: Freese & Valeriote’s theorem For
Page 67 and 68: Freese & Valeriote’s theorem Proo
Page 69 and 70: Open Problem 2. Are the following e
Page 71 and 72: Open Problem 2. Are the following e
Page 73 and 74: Surprisingly, the previous problems
Page 75 and 76: Variety Membership Problem (VAR-MEM
Page 77 and 78: What is the “real” complexity o
Page 79 and 80: The Equivalence of Terms problem (E
Page 81 and 82: Obviously INEQUIV -TERM is in NP. (
Page 83 and 84: Obviously INEQUIV -TERM is in NP. (
Page 85 and 86: But WAIT!!!! There’s more!!!! For
Page 91 and 92: There are a huge number of publicat
Page 93 and 94: There are a huge number of publicat
Page 95 and 96: An outrageous scandal Theorem (G. H
Page 97 and 98: Equivalence of Terms Problem (corre
Page 99 and 100: Equivalence of Terms Problem (corre
Page 101 and 102: Two problems for relational structu
Page 103: Fix a finite relational structure B
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Universal Algebra and Computational Complexity Lecture 3

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