the algebraic method for constraint satisfaction problemslac2018/... · the algebraic method for...
TRANSCRIPT
![Page 1: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/1.jpg)
The algebraic method for Constraint SatisfactionProblems
LAC 2018Institute of Mathematics for Industry, Kyushu University and La Trobe
University
Marcel Jackson
![Page 2: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/2.jpg)
Constraints and satisfaction
ConstraintA tuple of variables, and a target relation on some domain
Constraint satisfaction problemGiven some constraints, can they be satisfied?
![Page 3: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/3.jpg)
3SAT
Conjunction of clauses:
(x1 ∨ x2 ∨ x3) ∧ (¬x1 ∨ x2 ∨ ¬x3) ∧ (x1 ∧ ¬x2 ∧ ¬x4) ∧ . . .
Can the instance be satisfied?
![Page 4: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/4.jpg)
3SAT
Conjunction of clauses:
(x1 ∨ x2 ∨ x3) ∧ (¬x1 ∨ x2 ∨ ¬x3) ∧ (x1 ∧ ¬x2 ∧ ¬x4) ∧ . . .
Can the instance be satisfied?. The quintessential NP-complete classic
a classic catch by John Dyson 1981
![Page 5: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/5.jpg)
3SAT
Conjunction of clauses:
(x1 ∨ x2 ∨ x3) ∧ (¬x1 ∨ x2 ∨ ¬x3) ∧ (x1 ∧ ¬x2 ∧ ¬x4) ∧ . . .
Can the instance be satisfied?
As a CSPEach clause is a constraint:. Clause (¬x1 ∨ x2 ∨ ¬x3) means (x1, x2, x3) must lie in
{000,001,010,011,100,101,110,111}
![Page 6: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/6.jpg)
Not-all-equal 3SAT
Conjunction of clauses:
(x1 ∨ x2 ∨ x3) ∧ (¬x1 ∨ x2 ∨ ¬x3) ∧ (x1 ∧ ¬x2 ∧ ¬x4) ∧ . . .
Can the instance be satisfied with each clause containing a true literaland a false literal?
![Page 7: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/7.jpg)
Not-all-equal 3SAT
Conjunction of clauses:
(x1 ∨ x2 ∨ x3) ∧ (¬x1 ∨ x2 ∨ ¬x3) ∧ (x1 ∧ ¬x2 ∧ ¬x4) ∧ . . .
Can the instance be satisfied with each clause containing a true literaland a false literal?. Another well-known NP-complete classic.
A Warrick Capper classic
![Page 8: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/8.jpg)
Not-all-equal 3SAT
Conjunction of clauses:
(x1 ∨ x2 ∨ x3) ∧ (¬x1 ∨ x2 ∨ ¬x3) ∧ (x1 ∧ ¬x2 ∧ ¬x4) ∧ . . .
Can the instance be satisfied with each clause containing a true literaland a false literal?
As a CSPEach clause is a constraint:. Clause (¬x1 ∨ x2 ∨ ¬x3) means (x1, x2, x3) must lie in
{000,001,010,011,100,101,110,111}
![Page 9: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/9.jpg)
Solvability of linear equations
A system of equations over Z2:
x1 + x2 +x4 = 1x2 + x3 +x4 = 1
x1 + x3 +x4 = 0x1 +x4 = 1
![Page 10: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/10.jpg)
Solvability of linear equations
A system of equations over Z2:
x1 + x2 +x4 = 1x2 + x3 +x4 = 1
x1 + x3 +x4 = 0x1 +x4 = 1
. Easily solved in polynomial time using Gaussian elimination. Ithas its own complexity class ⊕L (“parity L”)
![Page 11: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/11.jpg)
Solvability of linear equations
A system of equations over Z2:
x1 + x2 +x4 = 1x2 + x3 +x4 = 1
x1 + x3 +x4 = 0x1 +x4 = 1
As a CSPEach equation is a constraint:. Equation x2 + x3 + x4 = 1 means (x2, x3, x4) is constrained to be in
{100,010,001,111}
![Page 12: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/12.jpg)
Directed graph unreachability
A directed graph, a “start” vertex s and a “finish” vertex t . Is there nodirected path from s to t?
![Page 13: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/13.jpg)
Directed graph unreachability
A directed graph, a “start” vertex s and a “finish” vertex t . Is there nodirected path from s to t?. Easily solved in polynomial time (and nondeterministic logspace).. A fundamental computational problem in computational complexity
![Page 14: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/14.jpg)
Directed graph unreachability
A directed graph, a “start” vertex s and a “finish” vertex t . Is there nodirected path from s to t?
As a CSPEach edge is a constraint:. (u, v) means (u, v) is constrained to be in {00,01,11} (that is, ≤
on 0,1)
![Page 15: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/15.jpg)
Directed graph unreachability
A directed graph, a “start” vertex s and a “finish” vertex t . Is there nodirected path from s to t?
As a CSPEach edge is a constraint:. (u, v) means (u, v) is constrained to be in {00,01,11} (that is, ≤
on 0,1)AND:. s is constrained to be 1 while t is constrained to be 0
![Page 16: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/16.jpg)
Schaefer’s Theorem
So far, all these problems have domain 0,1.
Schaefer’s Theorem (1979)A Boolean satisfiability problem is either solvable in P or NP-complete
![Page 17: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/17.jpg)
Graph colouring
Given a graph G = (V ,E), can we colour the vertices V by {0,1,2} sothat adjacent vertices have different colours?
![Page 18: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/18.jpg)
Graph colouring
Given a graph G = (V ,E), can we colour the vertices V by {0,1,2} sothat adjacent vertices have different colours?. Yet another classic NP-complete problem
![Page 19: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/19.jpg)
Graph colouring
Given a graph G = (V ,E), can we colour the vertices V by {0,1,2} sothat adjacent vertices have different colours?
As a CSPEach edge is a constraint:. (u, v) means (u, v) is constrained to be in {01,10,02,20,12,21}
(the 6= relation on {0,1,2})
![Page 20: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/20.jpg)
Graph colouring
Given a graph G = (V ,E), can we colour the vertices V by {0,1,2} sothat adjacent vertices have different colours?
As a CSPEach edge is a constraint:. (u, v) means (u, v) is constrained to be in {01,10,02,20,12,21}
(the 6= relation on {0,1,2})
The target domain and relations are fixed: “template”
![Page 21: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/21.jpg)
Graph colouring
Given a graph G = (V ,E), can we colour the vertices V by {0,1,2} sothat adjacent vertices have different colours?
As a CSPEach edge is a constraint:. (u, v) means (u, v) is constrained to be in {01,10,02,20,12,21}
(the 6= relation on {0,1,2})
The target domain and relations are fixed: “template”
Database exampleConjunctive database queries (the database is the template, the querythe instance)
![Page 22: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/22.jpg)
Ladner’s Theorem versus CSPs
Ladner’s TheoremIf P 6= NP then there are problems in NP\P that are not NP-complete.
. . . but in practice there seem to be few natural problems that appear tohave this intermediate status
Feder and Vardi (1994)(Roughly) there is a largest logically definable subclass of NP in whichLadner’s Theorem hold
![Page 23: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/23.jpg)
Ladner’s Theorem versus CSPs
Ladner’s TheoremIf P 6= NP then there are problems in NP\P that are not NP-complete.
Feder and Vardi (1994)(Roughly) there is a largest logically definable subclass of NP in whichLadner’s Theorem might not hold
![Page 24: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/24.jpg)
Ladner’s Theorem versus CSPs
Ladner’s TheoremIf P 6= NP then there are problems in NP\P that are not NP-complete.
Feder and Vardi (1994)(Roughly) there is a largest logically definable subclass of NP in whichLadner’s Theorem might not hold,. it’s the fixed finite template CSPs!
![Page 25: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/25.jpg)
Ladner’s Theorem versus CSPs
Ladner’s TheoremIf P 6= NP then there are problems in NP\P that are not NP-complete.
Feder and Vardi (1994)(Roughly) there is a largest logically definable subclass of NP in whichLadner’s Theorem might not hold,. it’s the fixed finite template CSPs!
(very roughly)
![Page 26: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/26.jpg)
Ladner’s Theorem versus CSPs
Ladner’s TheoremIf P 6= NP then there are problems in NP\P that are not NP-complete.
Feder and Vardi (1994)(Roughly) there is a largest logically definable subclass of NP in whichLadner’s Theorem might not hold,. it’s the fixed finite template CSPs!. Conjecture: Ladner’s Theorem fails for this class.
![Page 27: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/27.jpg)
Ladner’s Theorem versus CSPs
Ladner’s TheoremIf P 6= NP then there are problems in NP\P that are not NP-complete.
Feder and Vardi (1994)(Roughly) there is a largest logically definable subclass of NP in whichLadner’s Theorem might not hold,. it’s the fixed finite template CSPs!. Conjecture: Ladner’s Theorem fails for this class.
Bulatov/Zhuk (joint best paper award, FOCS 2017)A fixed template CSP is either solvable in P or is NP-complete.
![Page 28: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/28.jpg)
Ladner’s Theorem versus CSPs
Ladner’s TheoremIf P 6= NP then there are problems in NP\P that are not NP-complete.
Feder and Vardi (1994)(Roughly) there is a largest logically definable subclass of NP in whichLadner’s Theorem might not hold,. it’s the fixed finite template CSPs!. Conjecture: Ladner’s Theorem fails for this class.
Bulatov/Zhuk (joint best paper award, FOCS 2017)A fixed template CSP is either solvable in P or is NP-complete.
. they give a structural characterisation of hardness for an enormousclass of natural problems of interest. . .
![Page 29: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/29.jpg)
Goal
![Page 30: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/30.jpg)
Goal
![Page 31: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/31.jpg)
Goal
. Give some sort of appreciation to the background mathematicsunderlying the Bulatov/Zhuk result and proof
![Page 32: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/32.jpg)
Goal
. Give some sort of appreciation to the background mathematicsunderlying the Bulatov/Zhuk result and proof
. as well as how this approach and result can be used to achieveother complexity-theoretic classifications
![Page 33: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/33.jpg)
The CSP Dichotomy Theorem
Bulatov/Zhuk 2017A fixed template CSP is solvable in P if it has a “cyclic polymorphism”and is NP-complete otherwise.
![Page 34: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/34.jpg)
The CSP Dichotomy Theorem
Bulatov/Zhuk 2017A fixed template CSP is solvable in P if it has a “cyclic polymorphism”and is NP-complete otherwise.
. to be explained in due course
![Page 35: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/35.jpg)
The CSP Dichotomy Theorem
Bulatov/Zhuk 2017A fixed template CSP is solvable in P if it has a “cyclic polymorphism”and is NP-complete otherwise.
![Page 36: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/36.jpg)
The CSP Dichotomy Theorem
Bulatov/Zhuk 2017A fixed template CSP is solvable in P if it has a “cyclic polymorphism”and is NP-complete otherwise.
. the “easy” part
![Page 37: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/37.jpg)
The CSP Dichotomy Theorem
Bulatov/Zhuk 2017A fixed template CSP is solvable in P if it has a “cyclic polymorphism”and is NP-complete otherwise.
. the hard part
![Page 38: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/38.jpg)
The CSP Dichotomy Theorem
Bulatov/Zhuk 2017A fixed template CSP is solvable in P if it has a “cyclic polymorphism”and is NP-complete otherwise.
. the hard part
it’s really hard
![Page 39: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/39.jpg)
Polymorphism
template D
![Page 40: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/40.jpg)
Polymorphism
template D
AutomorphismAutomorphism: f : D→ D
![Page 41: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/41.jpg)
Polymorphism
template D
AutomorphismAutomorphism: f : D→ D
the set of automorphisms form a group action on D
![Page 42: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/42.jpg)
Polymorphism
template D
PolymorphismPolymorphism: f : Dn → D
![Page 43: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/43.jpg)
Polymorphism
template D
PolymorphismPolymorphism: f : Dn → D
the set of all polymorphisms forms an exotic algebra on D
![Page 44: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/44.jpg)
Cyclic
Bulatov/Zhuk 2017A fixed template CSP is solvable in P if it has a cyclic polymorphismand is NP-complete otherwise.
n-ary cyclic polymorphism
∀x1 . . . ∀xn c(x1, x2, . . . , xn) = c(x2, . . . , xn, x1)
Behind the scenes (Barto, Kozik 2012 after Taylor 1977)A finite algebra has a cyclic term if and only if its variety contains noalgebras with essentially trivial term operations (projections)
![Page 45: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/45.jpg)
Cyclic
Bulatov/Zhuk 2017A fixed template CSP is solvable in P if it has a cyclic polymorphismand is NP-complete otherwise.
n-ary cyclic polymorphism
∀x1 . . . ∀xn c(x1, x2, . . . , xn) = c(x2, . . . , xn, x1)
Behind the scenes (Barto, Kozik 2012 after Taylor 1977)A finite algebra has a cyclic term if and only if its variety contains noalgebras with essentially trivial term operations (projections)
![Page 46: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/46.jpg)
Cyclic
Bulatov/Zhuk 2017A fixed template CSP is solvable in P if it has a cyclic polymorphismand is NP-complete otherwise.
n-ary cyclic polymorphism
∀x1 . . . ∀xn c(x1, x2, . . . , xn) = c(x2, . . . , xn, x1)
Behind the scenes (Barto, Kozik 2012 after Taylor 1977)A finite algebra has a cyclic term if and only if its variety contains noalgebras with essentially trivial term operations (projections)
![Page 47: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/47.jpg)
Cyclic
Bulatov/Zhuk 2017A fixed template CSP is solvable in P if it has a cyclic polymorphismand is NP-complete otherwise.
n-ary cyclic polymorphism
∀x1 . . . ∀xn c(x1, x2, . . . , xn) = c(x2, . . . , xn, x1)
Behind the scenes (Barto, Kozik 2012 after Taylor 1977)A finite algebra has a cyclic term if and only if its variety contains noalgebras with essentially trivial term operations (projections)
. iff (roughly) the template relations can logically define the 3SATternary relations by way of primitive positive formulæ
![Page 48: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/48.jpg)
Cyclic
Bulatov/Zhuk 2017A fixed template CSP is solvable in P if it has a cyclic polymorphismand is NP-complete otherwise.
n-ary cyclic polymorphism
∀x1 . . . ∀xn c(x1, x2, . . . , xn) = c(x2, . . . , xn, x1)
Behind the scenes (Barto, Kozik 2012 after Taylor 1977)A finite algebra has a cyclic term if and only if its variety contains noalgebras with essentially trivial term operations (projections)
. iff it has cyclic terms of all prime arities greater than the size of thealgebra
![Page 49: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/49.jpg)
Example: Directed graph unreachability
R = 〈{0,1};≤, , 〉
Cyclic polymorphismThe operations of meet ∧ and join ∨ on 0,1 are cyclic polymorphisms.
![Page 50: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/50.jpg)
Example: Directed graph unreachability
R = 〈{0,1};≤, , 〉
Cyclic polymorphismThe operations of meet ∧ and join ∨ on 0,1 are cyclic polymorphisms.
![Page 51: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/51.jpg)
Example: Directed graph unreachability
R = 〈{0,1};≤, , 〉
Cyclic polymorphismThe operations of meet ∧ and join ∨ on 0,1 are cyclic polymorphisms.
if. . .
a ∧ b = c
≤ and ≤
a′ ∧ b′ = c′
![Page 52: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/52.jpg)
Example: Directed graph unreachability
R = 〈{0,1};≤, , 〉
Cyclic polymorphismThe operations of meet ∧ and join ∨ on 0,1 are cyclic polymorphisms.
and
a ∧ b = c
≤ and ≤
a′ ∧ b′ = c′
![Page 53: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/53.jpg)
Example: Directed graph unreachability
R = 〈{0,1};≤, , 〉
Cyclic polymorphismThe operations of meet ∧ and join ∨ on 0,1 are cyclic polymorphisms.
then. . .
a ∧ b = c
≤ and ≤ ≤
a′ ∧ b′ = c′
![Page 54: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/54.jpg)
Example: Directed graph unreachability
R = 〈{0,1};≤, , 〉
Cyclic polymorphismThe operations of meet ∧ and join ∨ on 0,1 are cyclic polymorphisms.
and obviously x1 ∧ x2 = x2 ∧ x1 (for all x1, x2 ∈ {0,1})
![Page 55: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/55.jpg)
Hell-Nešetril Dichotomy
Hell and Nešetril (1990)A finite graph has tractable CSP if it has a loop or is bipartite and isNP-complete otherwise
Modern proof by cyclic polymorphism. loop: trivial. bipartite: easy (logspace)
. now assume there is an odd circuit.an odd length circuit︷ ︸︸ ︷
u1 − u2 − u3 − · · · − up − u1Claim: if there is a cyclic polymorphism c, there is a loop
c( u1, u2, . . . up−1, up)
c( u2, u3, . . . up, u1)
![Page 56: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/56.jpg)
Hell-Nešetril Dichotomy
Hell and Nešetril (1990)A finite graph has tractable CSP if it has a loop or is bipartite and isNP-complete otherwise
Modern proof by cyclic polymorphism. loop: trivial
. bipartite: easy (logspace)
. now assume there is an odd circuit.an odd length circuit︷ ︸︸ ︷
u1 − u2 − u3 − · · · − up − u1Claim: if there is a cyclic polymorphism c, there is a loop
c( u1, u2, . . . up−1, up)
c( u2, u3, . . . up, u1)
![Page 57: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/57.jpg)
Hell-Nešetril Dichotomy
Hell and Nešetril (1990)A finite graph has tractable CSP if it has a loop or is bipartite and isNP-complete otherwise
Modern proof by cyclic polymorphism. loop: trivial. bipartite: easy (logspace)
. now assume there is an odd circuit.an odd length circuit︷ ︸︸ ︷
u1 − u2 − u3 − · · · − up − u1Claim: if there is a cyclic polymorphism c, there is a loop
c( u1, u2, . . . up−1, up)
c( u2, u3, . . . up, u1)
![Page 58: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/58.jpg)
Hell-Nešetril Dichotomy
Hell and Nešetril (1990)A finite graph has tractable CSP if it has a loop or is bipartite and isNP-complete otherwise
Modern proof by cyclic polymorphism. loop: trivial. bipartite: easy (logspace)
. now assume there is an odd circuit.an odd length circuit︷ ︸︸ ︷
u1 − u2 − u3 − · · · − up − u1
Claim: if there is a cyclic polymorphism c, there is a loop
c( u1, u2, . . . up−1, up)
c( u2, u3, . . . up, u1)
![Page 59: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/59.jpg)
Hell-Nešetril Dichotomy
Hell and Nešetril (1990)A finite graph has tractable CSP if it has a loop or is bipartite and isNP-complete otherwise
Modern proof by cyclic polymorphism. loop: trivial. bipartite: easy (logspace)
. now assume there is an odd circuit.an odd length circuit︷ ︸︸ ︷
u1 − u2 − u3 − · · · − up − u1Claim: if there is a cyclic polymorphism c, there is a loop
c( u1, u2, . . . up−1, up)
c( u2, u3, . . . up, u1)
![Page 60: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/60.jpg)
Hell-Nešetril Dichotomy
Hell and Nešetril (1990)A finite graph has tractable CSP if it has a loop or is bipartite and isNP-complete otherwise
Modern proof by cyclic polymorphism. loop: trivial. bipartite: easy (logspace)
. now assume there is an odd circuit.an odd length circuit︷ ︸︸ ︷
u1 − u2 − u3 − · · · − up − u1Claim: if there is a cyclic polymorphism c, there is a loop
c( u1, u2, . . . up−1, up)
c( u2, u3, . . . up, u1)
![Page 61: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/61.jpg)
Hell-Nešetril Dichotomy
Hell and Nešetril (1990)A finite graph has tractable CSP if it has a loop or is bipartite and isNP-complete otherwise
Modern proof by cyclic polymorphism. loop: trivial. bipartite: easy (logspace)
. now assume there is an odd circuit.an odd length circuit︷ ︸︸ ︷
u1 − u2 − u3 − · · · − up − u1Claim: if there is a cyclic polymorphism c, there is a loop
c( u1, u2, . . . up−1, up) = v
c( u2, u3, . . . up, u1) = v
![Page 62: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/62.jpg)
Hell-Nešetril Dichotomy
Hell and Nešetril (1990)A finite graph has tractable CSP if it has a loop or is bipartite and isNP-complete otherwise
Modern proof by cyclic polymorphism. loop: trivial. bipartite: easy (logspace)
. now assume there is an odd circuit.an odd length circuit︷ ︸︸ ︷
u1 − u2 − u3 − · · · − up − u1Claim: if there is a cyclic polymorphism c, there is a loop
c( u1, u2, . . . up−1, up) = v| | . . . | |
c( u2, u3, . . . up, u1) = v
![Page 63: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/63.jpg)
Hell-Nešetril Dichotomy
Hell and Nešetril (1990)A finite graph has tractable CSP if it has a loop or is bipartite and isNP-complete otherwise
Modern proof by cyclic polymorphism. loop: trivial. bipartite: easy (logspace)
. now assume there is an odd circuit.an odd length circuit︷ ︸︸ ︷
u1 − u2 − u3 − · · · − up − u1Claim: if there is a cyclic polymorphism c, there is a loop
c( u1, u2, . . . up−1, up) = v| | . . . | | |
c( u2, u3, . . . up, u1) = v
![Page 64: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/64.jpg)
Beyond satisfaction
Many other results follow from polymorphism analysisCounting CSPs (complexity of counting solutions)
valued CSPs (constraints have a cost. Minimise it.)approximation of CSPs (complexity of solving asymptotically mostconstraints)Universal Horn class membership. . .
![Page 65: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/65.jpg)
Beyond satisfaction
Many other results follow from polymorphism analysisCounting CSPs (complexity of counting solutions)valued CSPs (constraints have a cost. Minimise it.)
approximation of CSPs (complexity of solving asymptotically mostconstraints)Universal Horn class membership. . .
![Page 66: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/66.jpg)
Beyond satisfaction
Many other results follow from polymorphism analysisCounting CSPs (complexity of counting solutions)valued CSPs (constraints have a cost. Minimise it.)approximation of CSPs (complexity of solving asymptotically mostconstraints)
Universal Horn class membership. . .
![Page 67: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/67.jpg)
Beyond satisfaction
Many other results follow from polymorphism analysisCounting CSPs (complexity of counting solutions)valued CSPs (constraints have a cost. Minimise it.)approximation of CSPs (complexity of solving asymptotically mostconstraints)Universal Horn class membership. . .
![Page 68: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/68.jpg)
CSPs as homomorphism problems
Fixed template ACSP(A) is just the homomorphism problem for homomorphisms into A
instance︷︸︸︷B ?−→
template︷︸︸︷A
![Page 69: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/69.jpg)
CSPs as homomorphism problems
Fixed template ACSP(A) is just the homomorphism problem for homomorphisms into A
instance︷︸︸︷B ?−→
template︷︸︸︷A
a graph
(3-colouring is anedge-preservingfunction into K3)0 1
2
K3
![Page 70: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/70.jpg)
CSPs as homomorphism problems
Fixed template ACSP(A) is just the homomorphism problem for homomorphisms into A
instance︷︸︸︷B ?−→
template︷︸︸︷A
a graph
(3-colouring is anedge-preservingfunction into K3)0 1
2
K3
![Page 71: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/71.jpg)
CSPs as homomorphism problems
Fixed template ACSP(A) is just the homomorphism problem for homomorphisms into A
instance︷︸︸︷B ?−→
template︷︸︸︷A
a graph
(3-colouring is anedge-preservingfunction into K3)0 1
2
K3
![Page 72: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/72.jpg)
CSPs as homomorphism problems
Fixed template ACSP(A) is just the homomorphism problem for homomorphisms into A
instance︷︸︸︷B ?−→
template︷︸︸︷A
a graph
(3-colouring is anedge-preservingfunction into K3)0 1
2
K3
(3-colouring is anedge-preservingfunction into K3)
![Page 73: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/73.jpg)
Universal Horn
A non-constraint is implied if every solution maps it inside the targetrelation.
CSP to universal Horn. CSP(A): is there a homomorphism from B into A?. UHorn(A): are there no implied constraints?
Fix a finite structure A in signature R
B is an induced substructure of a direct power of AB satisfies the universal Horn sentences of Aif r ∈ R is a relation of arity n and (b1, . . . ,bn) /∈ rB then there is ahomomorphism φ from B to A with (φ(b1), . . . , φ(bn)) /∈ rA
B has no implied constraints relative to A
![Page 74: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/74.jpg)
Universal Horn
A non-constraint is implied if every solution maps it inside the targetrelation.
CSP to universal Horn. CSP(A): is there a homomorphism from B into A?. UHorn(A): are there no implied constraints?
Fix a finite structure A in signature R
B is an induced substructure of a direct power of A
B satisfies the universal Horn sentences of Aif r ∈ R is a relation of arity n and (b1, . . . ,bn) /∈ rB then there is ahomomorphism φ from B to A with (φ(b1), . . . , φ(bn)) /∈ rA
B has no implied constraints relative to A
![Page 75: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/75.jpg)
Universal Horn
A non-constraint is implied if every solution maps it inside the targetrelation.
CSP to universal Horn. CSP(A): is there a homomorphism from B into A?. UHorn(A): are there no implied constraints?
Fix a finite structure A in signature R
B is an induced substructure of a direct power of AB satisfies the universal Horn sentences of A
if r ∈ R is a relation of arity n and (b1, . . . ,bn) /∈ rB then there is ahomomorphism φ from B to A with (φ(b1), . . . , φ(bn)) /∈ rA
B has no implied constraints relative to A
![Page 76: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/76.jpg)
Universal Horn
A non-constraint is implied if every solution maps it inside the targetrelation.
CSP to universal Horn. CSP(A): is there a homomorphism from B into A?. UHorn(A): are there no implied constraints?
Fix a finite structure A in signature R
B is an induced substructure of a direct power of AB satisfies the universal Horn sentences of Aif r ∈ R is a relation of arity n and (b1, . . . ,bn) /∈ rB then there is ahomomorphism φ from B to A with (φ(b1), . . . , φ(bn)) /∈ rA
B has no implied constraints relative to A
![Page 77: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/77.jpg)
Universal Horn
A non-constraint is implied if every solution maps it inside the targetrelation.
CSP to universal Horn. CSP(A): is there a homomorphism from B into A?. UHorn(A): are there no implied constraints?
Fix a finite structure A in signature R
B is an induced substructure of a direct power of AB satisfies the universal Horn sentences of Aif r ∈ R is a relation of arity n and (b1, . . . ,bn) /∈ rB then there is ahomomorphism φ from B to A with (φ(b1), . . . , φ(bn)) /∈ rA
B has no implied constraints relative to A
![Page 78: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/78.jpg)
CSP and UHorn are different
![Page 79: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/79.jpg)
CSP and UHorn are different
ExampleCSP(R) is NL-complete, but UHorn(R) is first order definable
R = 〈{0,1};≤, , 〉
![Page 80: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/80.jpg)
CSP and UHorn are different
ExampleCSP(R) is NL-complete, but UHorn(R) is first order definable
R = 〈{0,1};≤, , 〉
![Page 81: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/81.jpg)
CSP and UHorn are different
ExampleCSP(G) is first order definable, but UHorn(G) is NP-complete
G
![Page 82: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/82.jpg)
CSP and UHorn are different
ExampleCSP(G) is first order definable, but UHorn(G) is NP-complete
G
![Page 83: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/83.jpg)
Main Result
Barto, Jackson, Ham (2017)UHorn(A) is solvable in P if A has an idempotent cyclic polymorphismand otherwise is NP-complete
(roughly)
![Page 84: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/84.jpg)
Main Result
Barto, Jackson, Ham (2017)UHorn(A) is solvable in P if A has an idempotent cyclic polymorphismand otherwise is NP-complete
(roughly)
![Page 85: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/85.jpg)
The easy part is the easiness part
Let Aconst be the result of adding singleton unary relations to A
If A has an idempotent cyclic polymorphism1 then by the Bulatov/Zhuk Dichotomy Theorem, CSP(Aconst) is
tractable (the very hard part of their result)2 make multiple calls to this to solve membership in UHorn(A)
![Page 86: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/86.jpg)
The easy part is the easiness part
Let Aconst be the result of adding singleton unary relations to A
If A has an idempotent cyclic polymorphism
1 then by the Bulatov/Zhuk Dichotomy Theorem, CSP(Aconst) istractable (the very hard part of their result)
2 make multiple calls to this to solve membership in UHorn(A)
![Page 87: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/87.jpg)
The easy part is the easiness part
Let Aconst be the result of adding singleton unary relations to A
If A has an idempotent cyclic polymorphism1 then by the Bulatov/Zhuk Dichotomy Theorem, CSP(Aconst) is
tractable (the very hard part of their result)
2 make multiple calls to this to solve membership in UHorn(A)
![Page 88: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/88.jpg)
The easy part is the easiness part
Let Aconst be the result of adding singleton unary relations to A
If A has an idempotent cyclic polymorphism1 then by the Bulatov/Zhuk Dichotomy Theorem, CSP(Aconst) is
tractable (the very hard part of their result)2 make multiple calls to this to solve membership in UHorn(A)
![Page 89: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/89.jpg)
The hard part is the hardness part
If A has no idempotent cyclic polymorphism
1 Then Aconst has no cyclic polymorphism2 Apply the ANT to Aconst
The All or Nothing Theorem (ANT); Ham, J, 2016(Nothing is easy part)
if D has no cyclic polymorphism then ∀k ∃` s.t. it is NP-hard todistinguish (i) from (ii):
I (i) B is every reasonable partial homomorphism on k elementsextends to a solution
I (ii) B has no homomorphism into A
reasonable: can be extended to any further ` elements. (ii) means No for UHorn(A). Argue (nontrivially) how (i) implies
YES for UHorn(A).
![Page 90: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/90.jpg)
The hard part is the hardness part
If A has no idempotent cyclic polymorphism1 Then Aconst has no cyclic polymorphism
2 Apply the ANT to Aconst
The All or Nothing Theorem (ANT); Ham, J, 2016(Nothing is easy part)
if D has no cyclic polymorphism then ∀k ∃` s.t. it is NP-hard todistinguish (i) from (ii):
I (i) B is every reasonable partial homomorphism on k elementsextends to a solution
I (ii) B has no homomorphism into A
reasonable: can be extended to any further ` elements. (ii) means No for UHorn(A). Argue (nontrivially) how (i) implies
YES for UHorn(A).
![Page 91: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/91.jpg)
The hard part is the hardness part
If A has no idempotent cyclic polymorphism1 Then Aconst has no cyclic polymorphism2 Apply the ANT to Aconst
The All or Nothing Theorem (ANT); Ham, J, 2016(Nothing is easy part)
if D has no cyclic polymorphism then ∀k ∃` s.t. it is NP-hard todistinguish (i) from (ii):
I (i) B is every reasonable partial homomorphism on k elementsextends to a solution
I (ii) B has no homomorphism into A
reasonable: can be extended to any further ` elements. (ii) means No for UHorn(A). Argue (nontrivially) how (i) implies
YES for UHorn(A).
![Page 92: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/92.jpg)
The hard part is the hardness part
If A has no idempotent cyclic polymorphism1 Then Aconst has no cyclic polymorphism2 Apply the ANT to Aconst
The All or Nothing Theorem (ANT); Ham, J, 2016(Nothing is easy part)
if D has no cyclic polymorphism then ∀k ∃` s.t. it is NP-hard todistinguish (i) from (ii):
I (i) B is every reasonable partial homomorphism on k elementsextends to a solution
I (ii) B has no homomorphism into A
reasonable: can be extended to any further ` elements. (ii) means No for UHorn(A). Argue (nontrivially) how (i) implies
YES for UHorn(A).
![Page 93: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/93.jpg)
The hard part is the hardness part
If A has no idempotent cyclic polymorphism1 Then Aconst has no cyclic polymorphism2 Apply the ANT to Aconst
The All or Nothing Theorem (ANT); Ham, J, 2016(Nothing is easy part)
if D has no cyclic polymorphism then ∀k ∃` s.t. it is NP-hard todistinguish (i) from (ii):
I (i) B is every reasonable partial homomorphism on k elementsextends to a solution
I (ii) B has no homomorphism into A
reasonable: can be extended to any further ` elements. (ii) means No for UHorn(A). Argue (nontrivially) how (i) implies
YES for UHorn(A).
![Page 94: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/94.jpg)
The hard part is the hardness part
If A has no idempotent cyclic polymorphism1 Then Aconst has no cyclic polymorphism2 Apply the ANT to Aconst
The All or Nothing Theorem (ANT); Ham, J, 2016(Nothing is easy part)
if D has no cyclic polymorphism then ∀k ∃` s.t. it is NP-hard todistinguish (i) from (ii):
I (i) B is every reasonable partial homomorphism on k elementsextends to a solution
I (ii) B has no homomorphism into A
reasonable: can be extended to any further ` elements
. (ii) means No for UHorn(A). Argue (nontrivially) how (i) impliesYES for UHorn(A).
![Page 95: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/95.jpg)
The hard part is the hardness part
If A has no idempotent cyclic polymorphism1 Then Aconst has no cyclic polymorphism2 Apply the ANT to Aconst
The All or Nothing Theorem (ANT); Ham, J, 2016(Nothing is easy part)
if D has no cyclic polymorphism then ∀k ∃` s.t. it is NP-hard todistinguish (i) from (ii):
I (i) B is every reasonable partial homomorphism on k elementsextends to a solution
I (ii) B has no homomorphism into A
reasonable: can be extended to any further ` elements. (ii) means No for UHorn(A). Argue (nontrivially) how (i) implies
YES for UHorn(A).
![Page 96: The algebraic method for Constraint Satisfaction Problemslac2018/... · The algebraic method for Constraint Satisfaction Problems LAC 2018 Institute of Mathematics for Industry, Kyushu](https://reader036.vdocuments.mx/reader036/viewer/2022070810/5f08daa17e708231d4240b08/html5/thumbnails/96.jpg)
ReferencesL. Barto, L. Ham and M. Jackson, Flexible satisfaction, in progress.arXiv1611.00886L. Barto and M. Kozik, Absorbing subalgebras, cyclic terms and the constraintsatisfaction problem, Logical Methods in Computer Science, 8/1:07 (2012), 1–26.A. Bulatov, A dichotomy theorem for nonuniform CSPs, FOCS 2017, pp319–330. arXiv:1703.03021T. Feder and M. Vardi, The computational structure of monotone monadic SNPand constraint satisfaction: a study through Datalog and group theory, SIAM J.Computing, 28 (1), 1998, 57–104.L. Ham and M. Jackson, All or Nothing: toward a promise problem dichotomy forconstraint satisfaction problems, in Principles and practice of ConstraintProgramming, J.C. Beck (Ed.): CP 2017, LNCS 10416, pp. 139–156, 2017.P. Hell and J. Nešetril, On the complexity of H-coloring, J. Combin. Theory Ser.B 48(1) (1990), 92–110.T. J. Schaefer, The Complexity of Satisfiability Problems, STOC (1978), pp.216–226.W. Taylor, Varieties obeying homotopy laws, Can J. Math. 29 (1977), 498–527.D. Zhuk, A Proof of CSP Dichotomy Conjecture, FOCS 2017, pp. 331–342.arXiv:704.01914