advanced topics in ai - ist.tugraz.at filen-queens problem • positioning of n queens on a nxn...

Gerald Steinbauer

Institute for Software Technology

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Advanced Topics of AI – Constraint Satisfaction Problem

Gerald Steinbauer


Advanced Topics in AI - Constraint Satisfaction Problem -

Gerald Steinbauer


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Map Coloring

– Map Coloring Problem: assign colors to each region s.t. no

neighboring regions have the same color

– Representation as CSP:

• 𝑉 = {𝑊𝐴,𝑁𝑇, 𝑆𝐴, 𝑄,𝑁𝑆𝑊,𝑉, 𝑇}; 𝑑𝑣𝑖 = {𝑟𝑒𝑑, 𝑔𝑟𝑒𝑒𝑛, 𝑏𝑙𝑢𝑒};

• 𝐶 = {𝑊𝐴 ≠ 𝑁𝑇,𝑊𝐴 ≠ 𝑆𝐴,𝑁𝑇 ≠ 𝑆𝐴,𝑁𝑇 ≠ 𝑄, 𝑆𝐴 ≠ 𝑄,

𝑆𝐴 ≠ 𝑁𝑆𝑊, 𝑆𝐴 ≠ 𝑉,𝑄 ≠ 𝑁𝑆𝑊,𝑁𝑆𝑊 ≠ 𝑉}

[ArtInt2002]

Gerald Steinbauer


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Map Coloring

– example solution: • 𝑊𝐴 = 𝑟𝑒𝑑,𝑁𝑇 = 𝑔𝑟𝑒𝑒𝑛, 𝑆𝐴 = 𝑏𝑙𝑢𝑒, 𝑄 = 𝑟𝑒𝑑,𝑁𝑆𝑊 = 𝑔𝑟𝑒𝑒𝑛, 𝑉 = 𝑟𝑒𝑑, 𝑇 = 𝑟𝑒𝑑

[ArtInt2002]

Gerald Steinbauer


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N-Queens Problem

• Positioning of N queens on

a NxN board so that none

of the queens endangers

any of the other queens

an example solution for the

8-queens problem

Gerald Steinbauer


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Motivation

• many important and interesting problems can be

efficient and elegant expressed by • a set of variables with domains

• a set of constraints on that variables – constrain the domain of a

variable

• constraint satisfaction problem (CSP) deals with this

representation

• many real-world application domains • configuration

• task and resource allocation

• scheduling

• planning

• generation of test cases

Gerald Steinbauer


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Relations

• definition • a relation over sets 𝑋1, … , 𝑋𝑛 is a subset

• 𝑅 ⊆ 𝑋1 ×⋯× 𝑋𝑛 =: 𝑋𝑗1≤𝑗≤𝑛

• the number 𝑛 is referred to as arity of 𝑅

• an 𝑛-ary relation on a set 𝑋 is a subset • 𝑅 ⊆ 𝑋𝑛 ≔ 𝑋 ×⋯× 𝑋 (𝑛 times)

• since relations are sets, set-theoretical operations

(union, intersection, complement) can be applied to

relations as well

Gerald Steinbauer


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Constraints, Relations and Variables

• constraints can be expressed by relations that restrict

value assignments to variables

• consider variables 𝑥1, 𝑥2, 𝑥3 and relations 𝐵, 𝐶 defined

by: • 𝐵 = 𝑥, 𝑦, 𝑧 ∈ 0…3 3 𝑥 < 𝑦 < 𝑧

• 𝐶 = {(𝑥, 𝑦, 𝑧) ∈ 0…3 3|𝑥 > 𝑦 > 𝑧}

Gerald Steinbauer


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Relations over Variables

• let 𝑉 be a set of variables. for 𝑣 ∈ 𝑉 , let 𝑑𝑜𝑚(𝑣) be a

non-empty set (of values), called the domain of 𝑣

• definition: a relation over (pairwise distinct) variables

𝑣1, … , 𝑣𝑛 ∈ 𝑉 is a pair 𝑅𝑣1,…,𝑣𝑛 ≔ ( 𝑣1, … , 𝑣𝑛 , 𝑅)

where 𝑅 is a relation over 𝑑𝑜𝑚 𝑣1 , … , 𝑑𝑜𝑚(𝑣𝑛)

• the sequence (𝑣1, … , 𝑣𝑛) is referred to as the scheme

(or: range), the set {𝑣1, … , 𝑣𝑛 } as the scope, and 𝑅 as

the graph of 𝑅𝑣1,…,𝑣𝑛

• we will not always distinguish between a relation over

variables and its graph (and between scope and

scheme), e. g., we write 𝑅𝑣1,…,𝑣𝑛 ⊆ 𝑑𝑜𝑚 𝑣1 ×⋯× 𝑑𝑜𝑚 𝑣𝑛

Gerald Steinbauer


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• a Constraint Satisfaction Problem (or CSP) is defined by • a set of variables 𝑉 = {𝑣1, 𝑣2, … , 𝑣𝑚}, and

• a set of constraints 𝐶 = {𝑐1, 𝑐2, … , 𝑐𝑛}

• each variable 𝑣𝑖 has a nonempty domain 𝑑𝑣𝑖 of

possible values

• each constraint 𝑐𝑗 involves some subset of the variables and specifies the allowable combinations of values for that subset

Constraint Satisfaction Problem (CSP) I

Gerald Steinbauer


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• a state of the problem is defined by an assignment of values to some or all of the variables

• an assignment that does not violate any constraints is called a consistent or legal assignment

• a complete assignment is one in which every variable is mentioned

• a solution to a CSP is a complete assignment that satisfies all the constraints

• let 𝑑 be the maxiumum domain size → 𝑂(𝑑 𝑚)

Constraint Satisfaction Problem (CSP) II

Gerald Steinbauer


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Map Coloring Interactive

Gerald Steinbauer


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Boolean Satisfiability

• problem instance (Boolean constraint network): • variables: (propositional) variables

• domains: truth values {0,1} for each variable

• constraints: defined by a propositional formula in these variables

• example: (𝑥1 ∨ ¬𝑥2 ∨ ¬𝑥3) ∧ (𝑥1 ∨ 𝑥2 ∨ 𝑥4)

• SAT as a constraint satisfaction problem: • given an arbitrary Boolean constraint network, is the network

solvable?

• SAT is NP-complete!

Gerald Steinbauer


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3-SAT

• 3-SAT: the problem of deciding if a propositional

formula in conjunctive normal form (CNF) with

clauses with at most 3 literals. 3-SAT is NP-complete

• 2-SAT: for this special problem (only 2 literals per

clause) exist polynomial algorithms

Gerald Steinbauer


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Computational Complexity

• in general the Constraint Satisfaction Problem (CSP)

with finite domains is NP-complete

• proof sketch: a CSP in general form can be reduced

to SAT problem

• special cases: some cases like CSPs with binary

variables and binary constraint can be solved faster

Gerald Steinbauer


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Constraint Network

• a constraint network is a triple 𝑁 ≔ 𝑉, 𝑑𝑜𝑚, 𝐶 where: • 𝑉 is a non-empty and finite set of variables

• 𝑑𝑜𝑚 is a function that assigns to each variable 𝑣 ∈ 𝑉 a non-empty

set 𝑑𝑜𝑚(𝑣) (𝑑𝑜𝑚(𝑣) is called the domain of 𝑣, elements of 𝑑𝑜𝑚(𝑣) are called values)

• 𝐶 is a set of relations over variables of 𝑉 (called constraints), i.e.,

each constraint is a relation 𝑅𝑥1,…,𝑥𝑚 over some scheme 𝑆 =

(𝑥1, … , 𝑥𝑚) of variables in 𝑉

Gerald Steinbauer


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Constraint Network

• if we assume an ordering of the variables in 𝑉 , we

can write networks more compactly:

• definition: a constraint network is a triple 𝑁 ≔𝑉,𝐷, 𝐶 where: • 𝑉 = 𝑣1, … , 𝑣𝑛 is a non-empty and finite sequence of variables

• 𝐷 = 𝐷1, … , 𝐷𝑛 is a sequence of domains for 𝑉 (𝐷𝑖 is the domain of

variable 𝑣𝑖)

• 𝐶 is a set of constraints 𝑅𝑥 where 𝑥 = 𝑣𝑖1 , … , 𝑣𝑖𝑚 is a scheme of

variables in 𝑉 and 𝑅 ⊆ 𝐷𝑖1 ×⋯× 𝐷𝑖1𝑚

Gerald Steinbauer


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Example – 4-Queens Problem

Gerald Steinbauer


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Solution of a Constraint Network

• definition: a solution of a constraint network

𝑁 = 𝑉,𝐷, 𝐶 i is a (variable) assignment

𝑎: 𝑉 → 𝐷𝑖𝑖:𝑣𝑖∈𝑉

• such that • 𝑎(𝑣𝑖) ∈ 𝐷𝑖, for each 𝑣𝑖 ∈ 𝑉

• (𝑎 𝑥1 , … , 𝑎 𝑥𝑚 ) ∈ 𝑅 for each 𝑅𝑥1,…,𝑥𝑚 constraints in 𝐶

• 𝑁 is called solvable (or: satisfiable) if 𝑁 has a solution

• 𝑆𝑜𝑙(𝑁) denotes the set of all solutions of 𝑁

Gerald Steinbauer


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Types of Constraints

• Unary Constraints – restrict the values of a single variable

– e.g., 𝑣1 ≠ 5

• Binary Constraints – binary constraints relate two variables

– e.g., 𝑣1 < 𝑣2

• Higher order constraints – involve three or more variables

– e.g., (𝑣1= 3 ⋁𝑣2 = 5)⋀𝑣3 > 4

• Extensional vs. Intensional – 𝑑𝑜𝑚 𝑣1 = {1,2}

– intensional: 𝑣1 = 𝑣2

– extensional: { 1,1 , (2,2)}

Gerald Steinbauer


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Global Constraints

• what are global constraints? • type of similar constraint relations . . .

• . . . differing in the number of variables

• semantically redundant: same constraint can be expressed by a

conjunction of simpler constraints

• similar structure: can be exploited by constraint solvers

• examples • sum constraint

• knapsack constraint

• element constraint

• all-different constraint

• cardinality constraints

Gerald Steinbauer


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all-different Constraint

• let 𝑣1, … , 𝑣𝑛 be variables each with a domain 𝐷𝑖 (1 ≤ 𝑖 ≤ 𝑛): 𝑎𝑙𝑙 − 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡(𝑣1, … , 𝑣𝑛) ≔

(𝑑1, … , 𝑑𝑛) ∈ 𝐷1 ×⋯× 𝐷𝑛|𝑑𝑖 ≠ 𝑑𝑗 for 𝑖 ≠ 𝑗

• the all-different constraint is a simple, but widely used

global constraint in constraint programming

• it allows for compact modeling of CSPs

Gerald Steinbauer


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Sum Constraint

• let 𝑣1, … , 𝑣𝑛, 𝑧 be variables with subsets of ℚ as

domain. for each 𝑣𝑖, let 𝑐𝑖 ∈ ℚ be some fixed scalar,

𝑐 = (𝑐1, … , 𝑐𝑛).

• definition: the sum constraint is defined as: 𝑠𝑢𝑚 𝑣1, … , 𝑣𝑛, 𝑧, 𝑐

≔ (𝑑1, … , 𝑑𝑛, 𝑑) ∈ 𝐷𝑖1≤𝑖≤𝑛

× 𝐷𝑧|𝑑 = 𝑐𝑖𝑑𝑖1≤𝑖≤𝑛

Gerald Steinbauer


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Global Cardinality Problem

• 𝑣1, … , 𝑣𝑛: assignment variables with 𝐷𝑣𝑖 ⊆ 𝑑1∗,… ,𝑑𝑚

∗

• 𝑐1, … , 𝑐𝑚: count variables with the set of integer as

domains

• the global cardinality constraint is defined as:

𝑔𝑐𝑐 𝑣1, … , 𝑣𝑛, 𝑐1, … , 𝑐𝑚 ≔(𝑑1, … , 𝑑𝑛, 𝑜1, … , 𝑜𝑚) ∈ 𝐷𝑣𝑖1≤𝑖≤𝑛 × 𝐷𝑐𝑗1≤𝑗≤𝑚 |

for each 𝑗, 𝑑𝑗∗ occurs 𝑖𝑛 𝑑1, … , 𝑑𝑛 exactly 𝑜𝑗 times

• can be considered a generalization of the all-different

constraint

Gerald Steinbauer


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Circuit Constraint

• let s = (𝑠1, … , 𝑠𝑛) be a permutation of 1,… , 𝑛 .

• define 𝐶𝑠 as the smallest set that contains 1 and with

each element 𝑖 also 𝑠𝑖

• (𝑠1, … , 𝑠𝑛) is called cyclic if 𝐶𝑠 = 1,… , 𝑛 .

• definition: let 𝑣1, … , 𝑣𝑛 be variables with domains

𝐷𝑖 = 1,… , 𝑛 (1 ≤ 𝑖 ≤ 𝑛). 𝑐𝑖𝑟𝑐𝑢𝑖𝑡(𝑣1, … , 𝑣𝑛) ≔𝑑1, … , 𝑑𝑛 ∈ 𝐷1 ×⋯× 𝐷𝑛 (𝑑1, … , 𝑑𝑛 is cyclic

• given an assignment a= (𝑑1, … , 𝑑𝑛) define

𝐴: 𝑣𝑖𝑣𝑑𝑖 𝑑𝑖 ∈ 𝐷𝑖 , 1 ≤ 𝑖 ≤ 𝑛

• then, 𝑎 satisfies 𝑐𝑖𝑟𝑐𝑢𝑖𝑡(𝑣1, … , 𝑣𝑛) iff (𝑉, 𝐴) is a

directed cycle (without proper sub-cycles)

Gerald Steinbauer


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Solving CSP

• basic concept • solving using search

• goal • find an assignment for the variables which is consistent and

complete

• basic approach • select variable and assign a corresponding value

• test consistency between instantiated variables and constraints

• backtrack in the case of an inconsistency

• search with backtracking is a complete search

algorithm • every existing solution is found

Gerald Steinbauer


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State Spaces

• a state space is a 4-tuple 𝒮 = 𝑆, 𝑠0, 𝑆∗, 𝑂 where • 𝑆 is a finite set of states

• 𝑠0 ∈ 𝑆 is the initial state

• 𝑆∗ ⊆ 𝑆 is the set of goal states

• 𝑂 is a finite set of operators, where each operator 𝑜 ∈ 𝑂 is a partial

function on 𝑆, i.e. 𝑜: 𝑆′ → 𝑆 for some 𝑆′ ⊆ 𝑆

• we say that an operator 𝑜 is applicable in state 𝑠 iff 𝑜(𝑠) is defined

Gerald Steinbauer


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State Spaces for Constraint Networks

• states represent different partial variable assignments

• usually operators represent adding an assignment

• the state spaces for constraint networks usually have

two special properties • the search graphs are trees (i.e., there is exactly one path from the

initial state to any reachable search state)

• all solutions are at the same level of the tree

• due to these properties, variations of depth-first

search are usually the method of choice for solving

constraint networks

Gerald Steinbauer


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Backtracking

• backtracking: search systematically for consistent

partial instantiations in a depth-first manner • forward phase: extend the current partial solution by assigning a

consistent value to some new variable (if possible)

• backward phase: if no consistent instantiation for the current

variable exists, we return to the previous variable

Gerald Steinbauer


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Backtracking backtracking(𝑁,𝑎):

input: a constraint network 𝑁 = 𝑉,𝐷, 𝐶 and a

partial assignment 𝑎 of 𝑁

output: a solution of 𝑁 or “inconsistent”

if 𝑎 is inconsistent with 𝑁 then

return “inconsistent”

if 𝑎 is defined for all variables in 𝑉 then

return 𝑎

select some variable 𝑣𝑖 for which 𝑎 is not defined

for each value 𝑥 form 𝐷𝑖

𝑎′ ≔ 𝑎 ∪ {𝑣𝑖 ↦ 𝑥}

𝑎′′ ← 𝑏𝑎𝑐𝑘𝑡𝑟𝑎𝑐𝑘𝑖𝑛𝑔 𝑁, 𝑎′

if 𝑎′′ is not “inconsistent” then

return 𝑎′′


Gerald Steinbauer


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Complexity - Backtracking

• backtracking performs uninformed search

• let 𝑑 the maximum number of values in 𝑑𝑜𝑚 𝑣𝑖 𝑖, … , 𝑛

• in worst case we have to check 𝑂(𝑑𝑛) instantiations

• unfortunately: let 𝑒 be the number of constraints and 𝑟 be the maximum arity of the constraints

• a consistent check (lookup in one related constraints)

has 𝑂(𝑟 log 𝑑 )

• we need at most 𝑒 constraints leading to 𝑂(𝑒 𝑟 log 𝑑 )

• we have to check up to 𝑑 values for a selection

𝑂(𝑑 𝑒 𝑟 log 𝑑 )

Gerald Steinbauer


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Backtracking - Example

Gerald Steinbauer


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Improving Backtracking

• dead ends increases runtime

• we like to improve backtracking in order to avoid as

many as possible dead ends

• minimize the search space

Gerald Steinbauer


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Ordered Search Spaces

• let 𝑁 = 𝑉,𝐷, 𝐶 be a constraint network

• definition (variable ordering) • a variable ordering of 𝐶 is a permutation of the variable set 𝑉

• we write variable orderings in sequence notation: 𝑣1, … , 𝑣𝑛

• definition (ordered search space) • let σ = 𝑣1, … , 𝑣𝑛 a variable ordering of 𝐶

• the ordered search space of 𝐶 along ordering 𝜎 is the state

space obtained from the unordered search space of 𝐶 by

restricting each operator 𝑜𝑣𝑖=𝑎𝑖to states with 𝑠 = 𝑖 − 1

• in other words, in the initial state, only 𝑣1 can be assigned, then

only 𝑣2, then only 𝑣3, . . .

Gerald Steinbauer


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The Importance of Good Orderings

• all ordered search spaces for the same constraint

network contain the same set of solution states

• however, the total number of states can vary

dramatically between different orderings

• the size of a state space is a (rough) measure for the

hardness of finding a solution, so we are interested in

small search spaces

• one way of measuring the quality of a state space is

by counting the number of dead ends: the fewer, the

better

Gerald Steinbauer


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Variable Ordering Example

• three variables 𝑥, 𝑦, 𝑧 with domains 𝐷𝑥 = 2,3,4 , 𝐷𝑦 = 2,5,6 , 𝐷𝑧 = 2,3,5 and the constraint that the

assignment to 𝑧 has to integer divide the

assignments to 𝑥, 𝑦

Gerald Steinbauer


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Dynamic Variable Ordering

• common heuristic: fail-first

• always select a variable whose remaining domain

has a minimal number of elements.

• intuition: few subtrees small search space

• extreme case: only one value left no search

compare unit propagation in DPLL procedure

• should be combined with a constraint propagation

technique such as forward checking or arc

consistency

Gerald Steinbauer


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Look-Ahead and Lock-Back

• look-ahead: invoked when next variable or next value

is selected. for example: • which variable should be instantiated next? prefer variables that

impose tighter constraints on the rest of the search space

• which value should be chosen for the next variable? maximize

the number of options for future assignments

• look-back: invoked when the backtracking step is

performed after reaching a dead end. for example: • how deep should we backtrack? avoid irrelevant backtrack

points (by analyzing reasons for the dead end and jumping back to

the source of failure)

• how can we learn from dead ends? record reasons for dead

ends as new constraints so that the same inconsistencies can be

avoided at later stages of the search

Gerald Steinbauer


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Backtracking with Look-Ahead look-ahead(𝑁,𝑎):

input: a constraint network 𝑁 = 𝑉,𝐷, 𝐶 and a partial assignment 𝑎 of 𝑁

output: a solution of 𝑁 or “inconsistent”

selectValue(𝑣𝑖 , 𝑎, 𝑁): procedure that selects and deletes a consistent value

𝑥𝑖 ∈ 𝐷𝑖; side-effect: 𝑁 is refined; returns 0, if all 𝑎 ∪ {𝑣𝑖 ↦ 𝑥} are inconsistent

if 𝑎 is inconsistent with 𝑁 then


if 𝑎 is defined for all variables in 𝑉 then

return 𝑎

select a variable 𝑣𝑖 for which 𝑎 is not defined

𝑁′ ← 𝑁,𝐷′𝑖 = 𝐷𝑖

while 𝐷′𝑖 is not empty

𝑥,𝑁′ ← 𝑠𝑒𝑙𝑒𝑐𝑡𝑉𝑎𝑙𝑢𝑒 𝑣𝑖 , 𝑎, 𝑁′

if 𝑥 ≠ 0

𝑎′ ← 𝑏𝑎𝑐𝑘𝑡𝑟𝑎𝑐𝑘𝑖𝑛𝑔 𝑁′, 𝑎 ∪ {𝑣𝑖 ↦ 𝑥}

if 𝑎′ is not “inconsistent” then

return 𝑎′


Gerald Steinbauer


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Forward Checking

selectValue-FC(𝑣𝑖 , 𝑎, 𝑁)

select and delete 𝑥 from 𝐷𝑖

for each 𝑣𝑗 sharing a constraint with 𝑣𝑖 for which 𝑎 is not defined

𝐷′𝑗 ← 𝐷𝑗

for each value 𝑦 ∈ 𝐷′𝑗

if not consistent(a ∪ {𝑣𝑖 ↦ 𝑥, 𝑣𝑗 ↦ 𝑦} )

remove 𝑦 from 𝐷′𝑗

if 𝐷′𝑗is empty then

return 0

else

𝐷𝑗 ← 𝐷′𝑗

return 𝑥

Gerald Steinbauer


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Look-Ahead

• forward checking is 𝑂(𝑒 𝑑2) in worst case where 𝑑 is

the cardinality of the largest domain and 𝑒 the

number of constraints

• remark • keeping the balance between pruning the search space and cost of

look-ahead

• forward checking offers good trade-off

Gerald Steinbauer


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Enforcing Consistency

• the more explicit and tight constraint networks are,

the more restricted is the search space of partial

solutions

• idea: infer new constraints without “removing" (by

methods called local consistency enforcing, bounded

consistency inference, constraint propagation)

• consistency-enforcing algorithms aim at assisting

search: How can we extend a given partial solution of

a small subnetwork to a partial solution of a larger

subnetwork?

Gerald Steinbauer


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Arc Consistency I

• let 𝑁 = 𝑉,𝐷, 𝐶 be a constraint network

• variable 𝑣𝑖 is arc-consistent relative to variable 𝑣𝑗 if for

each value 𝑎𝑖 ∈ 𝐷𝑖, there exists an 𝑎𝑗 ∈ 𝐷𝑗 with

(𝑎𝑖 , 𝑎𝑗) ∈ 𝑅𝑖𝑗 (in case that 𝑅𝑖𝑗 exists in 𝐶)

• an arc constraint 𝑅𝑖𝑗 is arc-consistent if 𝑣𝑖 is arc-

consistent relative to 𝑣𝑗 and 𝑣𝑗 is arc-consistent rel. to 𝑣𝑖

• a network 𝑁 is arc-consistent if all its arc constraints are

arc-consistent

Gerald Steinbauer


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Arc Consistency II

• lemma: checking whether a network 𝑁 = 𝑉,𝐷, 𝐶 is arc-

consistent requires at most 𝑒𝑑2 operations (where 𝑒 is

the number of its binary constraints and 𝑑 is an upper

bound of its domain sizes)

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Arc Consistency - Example

• consider a constraint network with two variables 𝑣1 and 𝑣2, domains 𝐷1 = 𝐷2 = {1,2,3}, and the binary

constraint expressed by 𝑣1 < 𝑣2.

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Arc Consistency III

• we can shrink the domain of 2 variables to achieve arc-

consistency

• if a value is not part of this solution to a subnetwork – it

will not be part of a global solution

• if arc-consistency is not achievable – no solution exists

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Domain Revising

revise(𝑣𝑖,𝑣𝑗 , 𝑅𝑖𝑗)

input: a network with two variables 𝑣𝑖,𝑣𝑗,

domains 𝐷𝑖,𝐷𝑗 and constraint 𝑅𝑖𝑗

output: a network with refined 𝐷𝑖 such that 𝑣𝑖 is

arc-consistent relative to 𝑣𝑗

for each 𝑎𝑖 ∈ 𝐷𝑖

if there is no 𝑎𝑗 ∈ 𝐷𝑗 with (𝑎𝑖 , 𝑎𝑗) ∈ 𝑅𝑖𝑗 then

remove 𝑎𝑖 from 𝐷𝑖

Gerald Steinbauer


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Domain Revising

• lemma: the complexity of revise is 𝑂(𝑑2), where 𝑑 is

an upper bound of the domain sizes.

• note: with a simple modification of the revise

algorithm one could improve to 𝑂(𝑡), where 𝑡 is the

maximal number of tuples occurring in one of the

binary constraints in the network.

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AC1

AC1(𝑁)

input: a constraint network 𝑁 = 𝑉,𝐷, 𝐶

output: 𝑁 arc-consistent, but equivalent to input

network

repeat

for each arc {𝑣𝑖 , 𝑣𝑗} with 𝑅𝑖𝑗 ∈ 𝐶

revise(𝑣𝑖 , 𝑣𝑗)

revise(𝑣𝑗 , 𝑣𝑖)

endfor

until no domain is changed

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Complexity AC1

• lemma: let 𝑁 be a constraint network with 𝑛 variables,

each with a domain of size ≤ 𝑘, and 𝑒 binary

constraints. applying AC1 on the network runs in time

𝑂(𝑒 𝑛 𝑘3).

• proof: one cycle through all binary constraints takes

𝑂(𝑒 𝑘2). In the worst case, one cycle just removes

one value from one domain. moreover, there are at

most 𝑛𝑘 values. This results in an upper bound of

𝑂(𝑒 𝑛 𝑘3). ∎

• note: if the input network is already arc-consistent,

then AC1 runs in time 𝑂(𝑒 𝑘2).

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AC3

• idea: no need to process all constraints if only a few domains have

changed. operate on a queue of constraints to be processed.

AC3(𝑁)

input: a constraint network 𝑁 = 𝑉,𝐷, 𝐶

output: 𝑁 arc-consistent, but equivalent to input network

for each pair 𝑣𝑖 , 𝑣𝑗 that occurs in a constraint 𝑅𝑖𝑗

𝑞𝑢𝑒𝑢𝑒 ← 𝑞𝑢𝑒𝑢𝑒 ∪ { 𝑣𝑖 , 𝑣𝑗 , (𝑣𝑗 , 𝑣𝑖)}

endfor

while 𝑞𝑢𝑒𝑢𝑒 is not empty

select and remove (𝑣𝑖 , 𝑣𝑗) from 𝑞𝑢𝑒𝑢𝑒

revise(𝑣𝑖 , 𝑣𝑗)

if revise(𝑣𝑖 , 𝑣𝑗) changed 𝐷𝑖 then

𝑞𝑢𝑒𝑢𝑒 ← 𝑞𝑢𝑒𝑢𝑒 ∪ { 𝑣𝑘 , 𝑣𝑖 |𝑘 ≠ 𝑖, 𝑘 ≠ 𝑗}

endwhile

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Complexity AC3

• lemma: let 𝑁 be a constraint network with 𝑛 variables,

each with a domain of size ≤ 𝑘, and 𝑒 binary

constraints. applying AC3 on the network runs in time

𝑂(𝑒 𝑘3).

• proof. consider a single constraint. each time, when it

is reintroduced into the queue, the domain of one of

its variables must have been changed. since there

are at most 2𝑘 values, AC3 processes each

constraint at most 2𝑘 times. Because we have

𝑒 constraints and processing of each is in time 𝑂(𝑘2), we obtain 𝑂(𝑒 𝑘3) . ∎

• note: If the input network is already arc-consistent,

then AC3 runs in 𝑂(𝑒 𝑘2).

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Remark AC

• note: enforcing arc consistency may already be

sufficient to show that a constraint network is

inconsistent one or more domains become empty!

• sometimes “enforcing arc consistency" is sufficient for

detecting inconsistent (unsolvable) networks; but . . .

• enforcing arc consistency is not complete for deciding

consistency of networks; because . . .

• inferences rely only on domain constraints and single

binary constraints defined on the domains

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Sudoku

Gerald Steinbauer


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Literature

• parts of the lecture are based on the “Constraint

Satisfaction Problems” lecture of the Research Group

Foundations of Artificial Intelligence of the University

of Freiburg (http://www.informatik.uni-freiburg.de/~ki/)

• Rina Dechter. Constraint Processing. The Morgan

Kaufmann Series in Artificial Intelligence. 2003.

http://www.informatik.uni-freiburg.de/~ki/



Gerald Steinbauer


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Thank You!

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