84
On the Minimal Constraint Satisfaction Problem Guillaume Escamocher Barry O’Sullivan October 14 th , 2015
On the MinimalConstraint Satisfaction Problem
Guillaume EscamocherBarry O’Sullivan
October 14th, 2015
Introduction
Operations and PropertiesMinimalizationConsistencies and Backbone Variables
ComplexityPrevious ResultsOur Results
Creating Minimal CSP instancesGeneratorEmpirical Tests
Conclusion
On the Minimal Constraint Satisfaction ProblemIntroduction 10/14/2015 Slide 2/69
ConstraintSatisfactionProblem
A CSP instance:
• A set V = {v1, . . . , vn} of n variables.
• A set {A1, . . . ,An} of n domains. For all i , Ai contains the dipossible values that vi can take.
• A set C of constraints specifying the k-tuples of valuesthat are allowed, and the ones that are forbidden, with kbeing the arity of the instance.
Question:Is there a solution, that is a set of n values, one in eachdomain, that satisfies all the constraints?
On the Minimal Constraint Satisfaction ProblemIntroduction 10/14/2015 Slide 3/69
Conventions
• We also consider trivial constraints. There are(nk
)constraints. The constraint hypergraph is complete. Eachk-tuple of values over k different domains is eitherallowed or forbidden.
• An allowed k-tuple of values is a compatible k-tuple. Aforbidden k-tuple of values is an incompatible k-tuple.
• A tuple of more than k values which does not contain anyincompatible k-tuple is a partial solution.
• A subset of a complete solution is completable.
On the Minimal Constraint Satisfaction ProblemIntroduction 10/14/2015 Slide 4/69
Example
• Arity is 2.
• 3 variables.
• 2 values in each domain.
• 3 constraints→ 12 couples of values:7 are compatible (lines), 5 are incompatible.
On the Minimal Constraint Satisfaction ProblemIntroduction 10/14/2015 Slide 5/69
Example
a
b
c d
e
f
Two solutions:
• (a, d , f )• (b, c , e)
On the Minimal Constraint Satisfaction ProblemIntroduction 10/14/2015 Slide 6/69
Minimal CSP (Montanari 1974)
A Minimal CSP instance:
• A CSP instance.
• Every compatible k-tuple is completable.
On the Minimal Constraint Satisfaction ProblemIntroduction 10/14/2015 Slide 7/69
NOT a Minimal CSP instance
a
b
c d
e
f
(b, f ) is not completable.
On the Minimal Constraint Satisfaction ProblemIntroduction 10/14/2015 Slide 8/69
AMinimal CSP instance
a
b
c d
e
f
Three solutions:
• (a, d , f ): completes (a, d), (a, f ) and (d , f ).• (b, c , e): completes (b, c), (b, e) and (c , e).• (b, c , f ): completes (b, f ) and (c , f ).
On the Minimal Constraint Satisfaction ProblemIntroduction 10/14/2015 Slide 9/69
What is the Problem?
Decision ProblemIs there a solution?
Search ProblemExhibit a solution.
On the Minimal Constraint Satisfaction ProblemIntroduction 10/14/2015 Slide 10/69
What is the Problem?
Decision ProblemIs there a solution?
Search ProblemExhibit a solution.
On the Minimal Constraint Satisfaction ProblemIntroduction 10/14/2015 Slide 10/69
When not using the Minimal CSP
• Customer: I want a car.
• Salesperson: No problem, we have a lot of cars. Some ofthem have automatic transmission, the others havemanual transmission. Which one do you prefer?
• Customer: Automatic.
• Salesperson: Some of our cars are green, the rest areorange. Which color do you prefer?
• Customer: Green.
• Salesperson: Sorry, we do not have any green car withautomatic transmission.
On the Minimal Constraint Satisfaction ProblemIntroduction 10/14/2015 Slide 11/69
When not using the Minimal CSP
• Customer: I want a car.
• Salesperson: No problem, we have a lot of cars. Some ofthem have automatic transmission, the others havemanual transmission. Which one do you prefer?
• Customer: Automatic.
• Salesperson: Some of our cars are green, the rest areorange. Which color do you prefer?
• Customer: Green.
• Salesperson: Sorry, we do not have any green car withautomatic transmission.
On the Minimal Constraint Satisfaction ProblemIntroduction 10/14/2015 Slide 11/69
When not using the Minimal CSP
• Customer: I want a car.
• Salesperson: No problem, we have a lot of cars. Some ofthem have automatic transmission, the others havemanual transmission. Which one do you prefer?
• Customer: Automatic.
• Salesperson: Some of our cars are green, the rest areorange. Which color do you prefer?
• Customer: Green.
• Salesperson: Sorry, we do not have any green car withautomatic transmission.
On the Minimal Constraint Satisfaction ProblemIntroduction 10/14/2015 Slide 11/69
When not using the Minimal CSP
• Customer: I want a car.
• Salesperson: No problem, we have a lot of cars. Some ofthem have automatic transmission, the others havemanual transmission. Which one do you prefer?
• Customer: Automatic.
• Salesperson: Some of our cars are green, the rest areorange. Which color do you prefer?
• Customer: Green.
• Salesperson: Sorry, we do not have any green car withautomatic transmission.
On the Minimal Constraint Satisfaction ProblemIntroduction 10/14/2015 Slide 11/69
When not using the Minimal CSP
• Customer: I want a car.
• Salesperson: No problem, we have a lot of cars. Some ofthem have automatic transmission, the others havemanual transmission. Which one do you prefer?
• Customer: Automatic.
• Salesperson: Some of our cars are green, the rest areorange. Which color do you prefer?
• Customer: Green.
• Salesperson: Sorry, we do not have any green car withautomatic transmission.
On the Minimal Constraint Satisfaction ProblemIntroduction 10/14/2015 Slide 11/69
When not using the Minimal CSP
• Customer: I want a car.
• Salesperson: No problem, we have a lot of cars. Some ofthem have automatic transmission, the others havemanual transmission. Which one do you prefer?
• Customer: Automatic.
• Salesperson: Some of our cars are green, the rest areorange. Which color do you prefer?
• Customer: Green.
• Salesperson: Sorry, we do not have any green car withautomatic transmission.
On the Minimal Constraint Satisfaction ProblemIntroduction 10/14/2015 Slide 11/69
Configuration Problems
• A car salesperson.
• Several cars to sell.
• Cars are characterized by their type of transmission,color, engine, price range.
• Several possibilities for each of the characteristics(automatic or manual, green or orange,. . . ).
• Customers can make choices. Each possible sequence ofchoices must be associated to at least one car.
On the Minimal Constraint Satisfaction ProblemIntroduction 10/14/2015 Slide 12/69
Configuration Problems
• A seller.
• Several items to sell.
• Several options characterize the items.
• Several choices for each characteristic.
• Each possible sequence of choices must be associated toat least one item.
Buying a product online
• Cars (make, color, engine,. . . )
• Flights (airline, destination, number of stops,. . . )
• Computers (brand, operating system, video card,. . . )
On the Minimal Constraint Satisfaction ProblemIntroduction 10/14/2015 Slide 13/69
Configuration Problems
• A seller.
• Several items to sell.
• Several options characterize the items.
• Several choices for each characteristic.
• Each possible sequence of choices must be associated toat least one item.
Buying a product online
• Cars (make, color, engine,. . . )
• Flights (airline, destination, number of stops,. . . )
• Computers (brand, operating system, video card,. . . )
On the Minimal Constraint Satisfaction ProblemIntroduction 10/14/2015 Slide 13/69
Configuration Problems
• A seller↔ A CSP instance
• Several items to sell↔ Several solutions to the instance
• Characteristics↔ Variables
• Several possible choices for each option↔ Several values for each variable
• Each possible sequence of choices must be associated toat least one item↔ Each partial solution must be completable
On the Minimal Constraint Satisfaction ProblemIntroduction 10/14/2015 Slide 14/69
When using the Minimal CSP
Queries that we can answer in polynomial time:
• Is there at least one green automatic car?Decision, O(d)
• At what time is the earliest flight from Cork to London?Optimization, O(d)
• Is there a computer such that(hard disk memory)/100+RAM≥16GB?Decision, O(d2)
No need to solve the entire instance, just look at oneconstraint.
On the Minimal Constraint Satisfaction ProblemIntroduction 10/14/2015 Slide 15/69
Conventions
• A CSP instance with empty domains is minimal.
• A CSP instance with only incompatible k-tuples is minimal.
We only study Minimal CSP instances with non-empty domainsand such that each value belongs to at least one compatiblek-tuple.
On the Minimal Constraint Satisfaction ProblemIntroduction 10/14/2015 Slide 16/69
Introduction
Operations and PropertiesMinimalizationConsistencies and Backbone Variables
ComplexityPrevious ResultsOur Results
Creating Minimal CSP instancesGeneratorEmpirical Tests
Conclusion
On the Minimal Constraint Satisfaction ProblemOperations and Properties 10/14/2015 Slide 17/69
Minimalizing a CSP instance
DefinitionTominimalize a CSP instance is to convert allnon-completable compatibilities into incompatibilities.
LemmaAll CSP instances are minimalizable. The result ofminimalization is unique.
Proof (Uniqueness)Converting a compatibility into an incompatibility does notchange the set of solutions.
On the Minimal Constraint Satisfaction ProblemMinimalization 10/14/2015 Slide 18/69
Minimalizing a CSP instance
DefinitionTominimalize a CSP instance is to convert allnon-completable compatibilities into incompatibilities.
LemmaAll CSP instances are minimalizable. The result ofminimalization is unique.
Proof (Uniqueness)Converting a compatibility into an incompatibility does notchange the set of solutions.
On the Minimal Constraint Satisfaction ProblemMinimalization 10/14/2015 Slide 18/69
Example
a
b
c d
e
f
Original instance
• Two solutions (a, d , f ) and (b, c , e).• (b, f ) is compatible and not completable.
On the Minimal Constraint Satisfaction ProblemMinimalization 10/14/2015 Slide 19/69
Example
a
b
c d
e
f
Minimalized instance
• Two solutions (a, d , f ) and (b, c , e).• Every compatible couple of values is completable.
On the Minimal Constraint Satisfaction ProblemMinimalization 10/14/2015 Slide 20/69
Complexity
PropositionMinimalizing a CSP instance is NP-hard.
Proof
• Reduction from the CSP.
• Get a CSP instance as input. Minimalize it.
• if something remains, return yes. Otherwise, return no.
CorollaryMinimalizing is NP-hard for every set of CSP instances that isNP-Complete in the general decision problem.
On the Minimal Constraint Satisfaction ProblemMinimalization 10/14/2015 Slide 21/69
Embedding in a Minimal CSP Instance
DefinitionA CSP instance I is embedded in another CSP instance I ′ if Iand I ′ share the same variables and domains, and everycompatiblity in I is a compatiblity in I ′.
LemmaEvery CSP instance is embedded in a Minimal CSP instance.The embedding is not unique, even for a given number ofcompatibilities.
Proof (Existence)Convert all incompatibilities into compatibilities.
On the Minimal Constraint Satisfaction ProblemMinimalization 10/14/2015 Slide 22/69
Example
a
b
c d
e
f
Original instance
• Two solutions (a, d , f ) and (b, c , e).• (b, f ) is compatible and not completable.
On the Minimal Constraint Satisfaction ProblemMinimalization 10/14/2015 Slide 23/69
Example 1
a
b
c d
e
f
Embedding instance 1
• Three solutions (a, d , f ), (b, c , e) and (b, c , f ).• Every compatible couple of values is completable.
On the Minimal Constraint Satisfaction ProblemMinimalization 10/14/2015 Slide 24/69
Example 2
a
b
c d
e
f
Embedding instance 2
• Three solutions (a, d , f ), (b, c , e) and (b, d , f ).• Every compatible couple of values is completable.
On the Minimal Constraint Satisfaction ProblemMinimalization 10/14/2015 Slide 25/69
(i , j)-consistency
Definition (Freuder 1985)A CSP instance I is (i , j)-consistent if for any set V1 of ivariables, and any set V2 of j variables such that V1 ∩ V2 = ∅,any compatible i -tuple on V1 can be extended to a compatible(i + j)-tuple on V1 ∪ V2.
Examples
• Arc consistency⇔ (1,1)-consistency
• Path consistency⇔ (2,1)-consistency
• Path inverse consistency⇔ (1,2)-consistency
On the Minimal Constraint Satisfaction ProblemConsistencies and Backbone Variables 10/14/2015 Slide 26/69
LemmaLet I be a k-ary Minimal CSP instance.∀i ≤ k , ∀j ≤ (n − i), I is (i , j)-consistent.
Proof/RemarkIt is actually an alternate definition of minimality.
On the Minimal Constraint Satisfaction ProblemConsistencies and Backbone Variables 10/14/2015 Slide 27/69
i-wise consistent
DefinitionA CSP instance I is i -wise consistent if for any setC0 = {c1, c2, . . . , ci} of i constraints of I , for any 1 ≤ j ≤ i , anycompatible k-tuple on cj can be extended to a compatibletuple on C0.
LemmaLet I be a Minimal CSP instance.∀i , I is i -wise consistent.
On the Minimal Constraint Satisfaction ProblemConsistencies and Backbone Variables 10/14/2015 Slide 28/69
Minimal CSP and Backbone Variables
DefinitionLet I be a CSP instance and v a variable of I . v is a backbonevariable if it takes the same value in all solutions to I .
LemmaIf I is a Minimal CSP instance, then the only backbonevariables of I are single-valued variables.
ProofIn a Minimal CSP, every value belongs to a solution.
On the Minimal Constraint Satisfaction ProblemConsistencies and Backbone Variables 10/14/2015 Slide 29/69
Introduction
Operations and PropertiesMinimalizationConsistencies and Backbone Variables
ComplexityPrevious ResultsOur Results
Creating Minimal CSP instancesGeneratorEmpirical Tests
Conclusion
On the Minimal Constraint Satisfaction ProblemComplexity 10/14/2015 Slide 30/69
The Minimal CSP
• Everything is in a solution.
• Consistent for many different notions of consistency.
• Can be used to quickly answer many queries.
⇒ Minimalization: very useful, albeit hard.(Not the main topic of the seminar)
The QuestionSince Minimal CSP instances look so easy, why study theircomplexity?
On the Minimal Constraint Satisfaction ProblemComplexity 10/14/2015 Slide 31/69
Because they are hard.
Theorem (Gottlob 2011)The Minimal Constraint Satisfaction Problem is NP-hard.
On the Minimal Constraint Satisfaction ProblemPrevious Results 10/14/2015 Slide 32/69
Proof for Binary Case
• Reduction from 3-SAT.
• I a SAT instance, n clauses, 3 literals in each clause.• v a variable→ v1, v2, v3, v4, v5 five variables.
• l a literal in a clause→ all ten combinations of lh ∨ li ∨ lj .• c a clause with 3 literals→ 103 = 1000 clauses with 3× 3 = 9 literals.
• SAT instance, 1000n clauses, 9 literals in each clause→ CSP instance, 1000n variables, 9 values in eachdomain.
• CSP instance is either minimal or unsatisfiable.
On the Minimal Constraint Satisfaction ProblemPrevious Results 10/14/2015 Slide 33/69
Proof for General Arity k
• Reduction from 3-SAT.
• SAT instance, n clauses, 3 literals in each clause.
→ CSP instance,(2k+1
k
)3× n variables, 3(k + 1) values in
each domain.• CSP instance is either minimal or unsatisfiable.
On the Minimal Constraint Satisfaction ProblemPrevious Results 10/14/2015 Slide 34/69
CorollaryMinimal CSP is NP-hard, even when bounding the size of thedomains by 3(k + 1).
On the Minimal Constraint Satisfaction ProblemPrevious Results 10/14/2015 Slide 35/69
Freuder 1991
TheoremFor binary CSP instances such that the treewidth of theconstraint graph is equal to h:(i , n − i)-consistency can be established in O((n × dh+1)i+1).
CorollaryMinimalizing binary CSP instances can be done inO((n × dh+1)3), with h the treewidth of the constraint graph.
On the Minimal Constraint Satisfaction ProblemPrevious Results 10/14/2015 Slide 36/69
Bounded Domain Size
Old Bound (Gottlob 2011)3(k + 1)→ 9
PropositionMinimal CSP is NP-hard, even when bounding the size of thedomains by d ≥ 3.
Proof
• Reduction fromMinimal CSP.
• One domain {a1, a2, . . . , am} of size m→ two domains {a1, a2, x} and {x , a3, . . . , am}of size 3 and m − 1.
On the Minimal Constraint Satisfaction ProblemOur Results 10/14/2015 Slide 37/69
Bounded Domain Size
Old Bound (Gottlob 2011)3(k + 1)→ 9
PropositionMinimal CSP is NP-hard, even when bounding the size of thedomains by d ≥ 3.
Proof
• Reduction fromMinimal CSP.
• One domain {a1, a2, . . . , am} of size m→ two domains {a1, a2, x} and {x , a3, . . . , am}of size 3 and m − 1.
On the Minimal Constraint Satisfaction ProblemOur Results 10/14/2015 Slide 37/69
Boolean Domains
PropositionBoolean Minimal CSP is NP-hard
, even when the arity isbounded by k ≥ 3.
Proof
• Reduction from 3-Minimal CSP.
• One domain {a1, a2, a3} of size 3→ three domains {a1, a1}, {a2, a2} and {a3, a3} of size 2.
On the Minimal Constraint Satisfaction ProblemOur Results 10/14/2015 Slide 38/69
Boolean Domains
PropositionBoolean Minimal CSP is NP-hard
, even when the arity isbounded by k ≥ 3.
Proof
• Reduction from 3-Minimal CSP.
• One domain {a1, a2, a3} of size 3→ three domains {a1, a1}, {a2, a2} and {a3, a3} of size 2.
On the Minimal Constraint Satisfaction ProblemOur Results 10/14/2015 Slide 38/69
Boolean Domains
PropositionBoolean Minimal CSP is NP-hard, even when the arity isbounded by k ≥ 3.
Proof
• Reduction from 3-Minimal CSP.
• One domain {a1, a2, a3} of size 3→ three domains {a1, a1}, {a2, a2} and {a3, a3} of size 2.
On the Minimal Constraint Satisfaction ProblemOur Results 10/14/2015 Slide 38/69
Binary Boolean Minimal CSP
PropositionBinary Boolean Minimal CSP is polynomial.
ProofSubset of Binary Boolean CSP, which is polynomial from 2-SAT.
On the Minimal Constraint Satisfaction ProblemOur Results 10/14/2015 Slide 39/69
The Minimal CSP Dichotomy Theoremk-ary Minimal CSP when the size of the domains is bounded byd is NP-hard if and only if (d ≥ 3 or (d = 2 and k ≥ 3)).
HHHHHHd
k2 ≥ 3
1 tractable tractable2 tractable NP-hard≥ 3 NP-hard NP-hard
On the Minimal Constraint Satisfaction ProblemOur Results 10/14/2015 Slide 40/69
Minimal CSP and general CSP
CorollaryMinimal CSP and the general CSP are NP-hard for the exactsame values of d and k .
On the Minimal Constraint Satisfaction ProblemOur Results 10/14/2015 Slide 41/69
Introduction
Operations and PropertiesMinimalizationConsistencies and Backbone Variables
ComplexityPrevious ResultsOur Results
Creating Minimal CSP instancesGeneratorEmpirical Tests
Conclusion
On the Minimal Constraint Satisfaction ProblemCreating Minimal CSP instances 10/14/2015 Slide 42/69
Theorem (Gottlob 2011)Minimal CSP is NP-hard.
ProofSAT instance, n clauses, 3 literals in each clause→ CSP instance, 1000n variables, 9 values in each domain
QuestionAre there hard minimal instances of reasonable size?
RemarkIt would be nice to have a way to efficiently generate random,reasonable, parameterizable, minimal instances.
On the Minimal Constraint Satisfaction ProblemGenerator 10/14/2015 Slide 43/69
Theorem (Gottlob 2011)Minimal CSP is NP-hard.
ProofSAT instance, n clauses, 3 literals in each clause→ CSP instance, 1000n variables, 9 values in each domain
QuestionAre there hard minimal instances of reasonable size?
RemarkIt would be nice to have a way to efficiently generate random,reasonable, parameterizable, minimal instances.
On the Minimal Constraint Satisfaction ProblemGenerator 10/14/2015 Slide 43/69
Theorem (Gottlob 2011)Minimal CSP is NP-hard.
ProofSAT instance, n clauses, 3 literals in each clause→ CSP instance, 1000n variables, 9 values in each domain
QuestionAre there hard minimal instances of reasonable size?
RemarkIt would be nice to have a way to efficiently generate random,reasonable, parameterizable, minimal instances.
On the Minimal Constraint Satisfaction ProblemGenerator 10/14/2015 Slide 43/69
DefinitionA k-ary CSP instance with n variables and d values in eachdomain is of size (d,n, k).
DefinitionThe tightness of a k-ary CSP instance is the number of itsincompatible k-tuples divided by its total number of k-tuples.
What we want
• Input d , n, k , t.• Output: A randomMinimal instance of size (d , n, k) andtightness t.
On the Minimal Constraint Satisfaction ProblemGenerator 10/14/2015 Slide 44/69
General CSPCan generate constraint by constraint, even tuple by tuple.Can edit an individual constraint, or even an individual tuple.
Minimal CSPModifying even a single tuple jeopardizes minimality.All constraints must be considered at once.
On the Minimal Constraint Satisfaction ProblemGenerator 10/14/2015 Slide 45/69
A Few Ideas
Reduction from other CSP instances
• Far too big instances!
• ‘‘Control’’ over the size, but not over the tightness.
Minimalization
• NP-hard⇒ not efficient.
• No control over the size or tightness.
On the Minimal Constraint Satisfaction ProblemGenerator 10/14/2015 Slide 46/69
A Few Ideas
Reduction from other CSP instances
• Far too big instances!
• ‘‘Control’’ over the size, but not over the tightness.
Minimalization
• NP-hard⇒ not efficient.
• No control over the size or tightness.
On the Minimal Constraint Satisfaction ProblemGenerator 10/14/2015 Slide 46/69
DefinitionA bare Minimal CSP instance is a CSP instance where allconstraints are equality constraints.
LemmaBare minimal instances are the minimal instances with thelowest possible tightness.
On the Minimal Constraint Satisfaction ProblemGenerator 10/14/2015 Slide 47/69
The Generator
Input: d , n, k integers, and t ∈ [0, 1].Output: Minimal CSP instance of size (d , n, k),
with a tightness (approximately) equal to t.
Algorithm
1. Generate a bare Minimal CSP instance I of size (d , n, k).2. ti ← tightness of I .3. While (ti > t)
3.1 Add solution to I .3.2 ti ← tightness of I .
4. Return I .
On the Minimal Constraint Satisfaction ProblemGenerator 10/14/2015 Slide 48/69
Issues
• The first n-tuple in the lexicographical order is a solution.
• For low values of t, the algorithm does not alwaysterminate.
On the Minimal Constraint Satisfaction ProblemGenerator 10/14/2015 Slide 49/69
The Improved Generator
Input: d , n, k integers, and t ∈ [0, 1].Output: Minimal CSP instance of size (d , n, k),
with a tightness (approximately) equal to t.
Algorithm
1. Generate a bare Minimal CSP instance I of size (d , n, k).2. Randomize the ordering of the values.
3. ti ← tightness of I .4. While (ti > t)
4.1 C ←most constrained constraint of I .4.2 A← random incompatible k-tuple from C .4.3 Add solution containing A to I .4.4 ti ← tightness of I .
5. Return I .
On the Minimal Constraint Satisfaction ProblemGenerator 10/14/2015 Slide 50/69
Final Tightness
Worst case
• Each iteration: one more compatibility in each constraint.
• Final deviation: 1dk .
Average case
• Each iteration: (current tightness) more compatibility ineach constraint.
• Final deviation: tdk .
On the Minimal Constraint Satisfaction ProblemGenerator 10/14/2015 Slide 51/69
QuestionIs the generator refined enough to detect hard Minimal CSPinstances?
Software
• Numberjack.• Solvers:
• Binary instances: Mistral.• 3+-ary instances: MiniSat.
• Stuff computed:• Is there a solution?• Number of nodes needed to find a solution.
⇒ Runtimes do not matter.
On the Minimal Constraint Satisfaction ProblemEmpirical Tests 10/14/2015 Slide 52/69
QuestionIs the generator refined enough to detect hard Minimal CSPinstances?
Software
• Numberjack.• Solvers:
• Binary instances: Mistral.• 3+-ary instances: MiniSat.
• Stuff computed:• Is there a solution?• Number of nodes needed to find a solution.
⇒ Runtimes do not matter.
On the Minimal Constraint Satisfaction ProblemEmpirical Tests 10/14/2015 Slide 52/69
0
10000
20000
30000
40000
50000
60000
70000
80000
90000
100000
0 50000 100000 150000 200000 250000 300000
Avera
ge s
earc
h e
ffort
(nodes)
Number of compatibilities
(a) Size (10, 76, k = 2)
On the Minimal Constraint Satisfaction ProblemEmpirical Tests 10/14/2015 Slide 53/69
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
50000
0 50000 100000 150000 200000 250000 300000
Avera
ge s
earc
h e
ffort
(nodes)
Number of compatibilities
(b) Size (15, 50, k = 2)
On the Minimal Constraint Satisfaction ProblemEmpirical Tests 10/14/2015 Slide 54/69
0
500
1000
1500
2000
2500
0 50000 100000 150000 200000 250000 300000
Avera
ge s
earc
h e
ffort
(nodes)
Number of compatibilities
(c) Size (27, 28, k = 2)
On the Minimal Constraint Satisfaction ProblemEmpirical Tests 10/14/2015 Slide 55/69
0
200
400
600
800
1000
1200
1400
0 50000 100000 150000 200000 250000 300000 350000 400000 450000
Avera
ge s
earc
h e
ffort
(nodes)
Number of compatibilities
(d) Size (6, 24, k = 3)
On the Minimal Constraint Satisfaction ProblemEmpirical Tests 10/14/2015 Slide 56/69
What Correlates Tightness and Hardness?
Phase Transitions
• CSP: number of satisfiable instances.
• Satisfiable CSP: number of backbone variables.
On the Minimal Constraint Satisfaction ProblemEmpirical Tests 10/14/2015 Slide 57/69
p-Step Instance
DefinitionA p-step instance of a CSP instance I is an instance obtainedfrom I after making p assignments and propagating(1, k − 1)-consistency after each assignment.
On the Minimal Constraint Satisfaction ProblemEmpirical Tests 10/14/2015 Slide 58/69
What Correlates Tightness and Hardness?
Phase Transitions
• CSP: number of satisfiable instances.
• Satisfiable CSP: number of backbone variables.
• Minimal CSP: number of satisfiable (k + 1)-step instances.
Why k + 1?All k-step instances are satisfiable.
On the Minimal Constraint Satisfaction ProblemEmpirical Tests 10/14/2015 Slide 59/69
0
10000
20000
30000
40000
50000
60000
70000
80000
90000
100000
0 50000 100000 150000 200000 250000 300000 0
20
40
60
80
100
Avera
ge s
earc
h e
ffort
(nodes)
Perc
enta
ge o
f satisfiable
(k+
1)-
ste
p insta
nces
Number of compatibilities
effortcompletability
(e) Size (10, 76, k = 2)
On the Minimal Constraint Satisfaction ProblemEmpirical Tests 10/14/2015 Slide 60/69
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
50000
0 50000 100000 150000 200000 250000 300000 0
20
40
60
80
100
Avera
ge s
earc
h e
ffort
(nodes)
Perc
enta
ge o
f satisfiable
(k+
1)-
ste
p insta
nces
Number of compatibilities
effortcompletability
(f) Size (15, 50, k = 2)
On the Minimal Constraint Satisfaction ProblemEmpirical Tests 10/14/2015 Slide 61/69
0
500
1000
1500
2000
2500
0 50000 100000 150000 200000 250000 300000 0
20
40
60
80
100
Avera
ge s
earc
h e
ffort
(nodes)
Perc
enta
ge o
f satisfiable
(k+
1)-
ste
p insta
nces
Number of compatibilities
effortcompletability
(g) Size (27, 28, k = 2)
On the Minimal Constraint Satisfaction ProblemEmpirical Tests 10/14/2015 Slide 62/69
0
200
400
600
800
1000
1200
1400
0 50000 100000 150000 200000 250000 300000 350000 400000 450000 0
20
40
60
80
100
Avera
ge s
earc
h e
ffort
(nodes)
Perc
enta
ge o
f satisfiable
(k+
1)-
ste
p insta
nces
Number of compatibilities
effortcompletability
(h) Size (6, 24, k = 3)
On the Minimal Constraint Satisfaction ProblemEmpirical Tests 10/14/2015 Slide 63/69
Conjecture
• In a Minimal CSP instance, the first k choices are free.
• The satisfiability of (k + 1)-step instances represents theprobability of making a right (k + 1)th choice.
• A lot of compatibilities means a large search space.
⇒ The hardest instances occur at low number of satisfiable(k + 1)-step instances and high number of compatibilities.
On the Minimal Constraint Satisfaction ProblemEmpirical Tests 10/14/2015 Slide 64/69
Introduction
Operations and PropertiesMinimalizationConsistencies and Backbone Variables
ComplexityPrevious ResultsOur Results
Creating Minimal CSP instancesGeneratorEmpirical Tests
Conclusion
On the Minimal Constraint Satisfaction ProblemConclusion 10/14/2015 Slide 65/69
Our Contributions: Complexity Theorem
The Minimal CSP Dichotomy Theoremk-ary Minimal CSP when the size of the domains is bounded byd is NP-hard if and only if (d ≥ 3 or (d = 2 and k ≥ 3)).
HHHH
HHdk
2 ≥ 3
1 tractable tractable2 tractable NP-hard≥ 3 NP-hard NP-hard
On the Minimal Constraint Satisfaction ProblemConclusion 10/14/2015 Slide 66/69
Our Contributions: Generator
Input: d , n, k integers, and t ∈ [0, 1].Output: Minimal CSP instance of size (d , n, k),
with a tightness equal to t ′ ∈ [t − 1dk , t].
Algorithm
1. Generate a bare Minimal CSP instance I of size (d , n, k).2. Randomize the ordering of the values.
3. ti ← tightness of I .4. While (ti > t)
4.1 C ←most constrained constraint of I .4.2 A← random incompatible k-tuple from C .4.3 Add solution containing A to I .4.4 ti ← tightness of I .
5. Return I .
On the Minimal Constraint Satisfaction ProblemConclusion 10/14/2015 Slide 67/69
(Very informal) Summary
• Minimal CSP definition: everything is part of a solution.
• It looks very easy, but it is actually hard.
On the Minimal Constraint Satisfaction ProblemConclusion 10/14/2015 Slide 68/69
Thank you very much for your attention!
On the Minimal Constraint Satisfaction ProblemConclusion 10/14/2015 Slide 69/69
