Introduction to Artificial Intelligence Local Search and Constraint Satisfaction

1
Introduction to Artificial IntelligenceLocal
SearchandConstraint Satisfaction

• Henry Kautz

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Local Search in Continuous Spaces
gradient
negative step to minimize f positive step to
maximize f
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Local Search in Discrete State Spaces

• state choose_start_state()
• while ! GoalTest(state) do
• state arg min h(s) s in Neighbors(state)
• end
• return state
• Terminology
• neighbors instead of children
• heuristic h(s) is the objective function, no
  need to be admissible
• No guarantee of finding a solution
• sometimes probabilistic guarantee
• Best goal-finding, not path-finding
• Many variations

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Local Search versus Systematic Search

• Systematic Search
• BFS, DFS, IDS, Best-First, A
• Keeps some history of visited nodes
• Always complete for finite search spaces, some
  versions complete for infinite spaces
• Good for building up solutions incrementally
• State partial solution
• Action extend partial solution

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Local Search versus Systematic Search

• Local Search
• Gradient descent, Greedy local search, Simulated
  Annealing, Genetic Algorithms
• Does not keep history of visited nodes
• Not complete. May be able to argue will terminate
  with high probability
• Good for iterative repair of candidate solutions
• State complete candidate solution that may not
  satisfy all constraints
• Action make a small change in the candidate
  solution

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N-Queens Problem
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N-Queens Systematic Search

• state choose_start_state()
• add state to Fringe
• while ! GoalTest(state) do
• choose state from Fringe according to h(state)
• Fringe Fringe U Children(state)
• end
• return state
• start empty board
• GoalTest N queens are on the board
• h (N number of queens on the board)
• children all ways of adding one queen without
  creating any attacks

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N-Queens Local Search, V1

• state choose_start_state()
• while ! GoalTest(state) do
• state arg min h(s) s in Neighbors(state)
• end
• return state
• start put down N queens randomly
• GoalTest Board has no attacking pairs
• h number of attacking pairs
• neighbors move one queen to a different square
  on the board

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N-Queens Local Search, V2

• state choose_start_state()
• while ! GoalTest(state) do
• state arg min h(s) s in Neighbors(state)
• end
• return state
• start put a queen on each square with 50
  probability
• GoalTest Board has N queens, no attacking pairs
• h (number of attacking pairs max(0, N -
  queens))
• neighbors add or delete one queen

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N Queens Demo
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States Where Greedy Search Must Succeed
objective function
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States Where Greedy Search Might Succeed
objective function
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Local Search Landscape
objective function
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Variations of Greedy Search

• Where to start?
• RANDOM STATE
• PRETTY GOOD STATE
• What to do when a local minimum is reached?
• STOP
• KEEP GOING
• Which neighbor to move to?
• BEST neighbor
• Any BETTER neighbor (Hill Climbing)
• How to make local search more robust?

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Restarts

• for run 1 to max_runs do
• state choose_start_state()
• flip 0
• while ! GoalTest(state) flip lt max_flips do
• state arg min h(s) s in Neighbors(state)
  
• end
• if GoalTest(state) return state
• end
• return FAIL

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Uphill Moves Random Noise

• state choose_start_state()
• while ! GoalTest(state) do
• with probability noise do
• state random member Neighbors(state)
• else
• state arg min h(s) s in
  Neighbors(state)
• end
• end
• return state

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Uphill Moves Simulated Annealing (Constant
Temperature)

• state start
• while ! GoalTest(state) do
• next random member Neighbors(state)
• deltaE h(next) h(state)
• if deltaE ? 0 then
• state next
• else
• with probability e-deltaE/temperature do
• state next
• end
• endif
• end
• return state

Book reverses, because is looking for max h state
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Uphill Moves Simulated Annealing (Geometric
Cooling Schedule)

• temperature start_temperature
• state choose_start_state()
• while ! GoalTest(state) do
• next random member Neighbors(state)
• deltaE h(next) h(state)
• if deltaE ? 0 then
• state next
• else
• with probability e-deltaE/temperature do
• state next
• end
• temperature cooling_rate temperature
• end
• return state

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Simulated Annealing

• For any finite problem with a fully-connected
  state space, will provably converge to optimum as
  length of schedule increases
• But fomal bound requires exponential search time
• In many practical applications, can solve
  problems with a faster, non-guaranteed schedule

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Other Local Search Strategies

• Tabu Search
• Keep a history of the last K visited states
• Revisiting a state on the history list is tabu
• Genetic algorithms
• Population set of K multiple search points
• Neighborhood population U mutations U
  crossovers
• Mutation random change in a state
• Crossovers random mix of assignments from two
  states
• Typically only a portion of neighbor is generated
• Search step new population K best members of
  neighborhood

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Myopic Local Search

• The local search methods we have discussed so far
  are myopic they only look at the immediate
  neighborhood of a single state at any one time
• Simple Parallelism run many searches in parallel
  with different random seeds
• Prob(Success) 1 Prob(Run Fails)k
• E.g. Prob(Run Fails) 90, k 10 ?
  Prob(Success) 65

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Multi-Point Local Search

• We can (sometimes) do better by considering
  several points simultaneously and exchanging
  information between the search points
• Two biological metaphors
• Genetic algorithms
• Swarm algorithms

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Genetic algorithms

• A successor state is generated by combining two
  parent states
  
• Start with k randomly generated states
  (population)
  
• A state is represented as a string over a finite
  alphabet (often a string of 0s and 1s)
  
• Evaluation function (fitness function). Depending
  on problem, may want to MAXIMIZE or MINIMIZE.
  
• Produce the next generation of states by
  selection, crossover, and mutation
  

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Example 8-Queens

• Fitness function number of non-attacking pairs
  of queens (min 0, max 8 7/2 28)
  
• 24/(24232011) 31
• 23/(24232011) 29 etc

Normalized Fitness
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The GA Cycle of Reproduction
children
reproduction
modification
modified children
parents
evaluation
population
evaluated children
deleted members
discard
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Population
population

• Chromosomes could be
• Bit strings
  (0101 ... 1100)
• Real numbers (43.2 -33.1 ...
  0.0 89.2)
• Permutations of element (E11 E3 E7 ... E1
  E15)
• Lists of rules (R1 R2 R3
  ... R22 R23)
• Program elements (genetic
  programming)
• ... any data structure ...

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Reproduction
children
reproduction
parents
population
Parents are selected at random with selection
chances biased in relation to chromosome
evaluations
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Chromosome Modification

• Modifications are stochastically triggered
• Operator types are
• Mutation
• Crossover (recombination)

children
modification
modified children
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Crossover mechanisms
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Evaluation

• The evaluator decodes a chromosome and assigns it
  a fitness measure
• The evaluator is the only link between a
  classical GA and the problem it is solving

modified children
evaluated children
evaluation
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Deletion
population

• Generational GAentire populations replaced with
  each iteration
• Steady-state GAa few members replaced each
  generation

discarded members
discard
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Example Traveling Salesman Problem

• Find a tour of a given set of cities so that
• each city is visited only once
• the total distance traveled is minimized

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Representation

• Representation is an ordered list of city
• numbers known as an order-based GA.
• 1) Berlin 3) Stuttgart 5) Cologne
  7) Dusseldorf
• 2) Munich 4) Wesbaden 6) Hanover 8)
  Breme
• CityList1 (3 5 7 2 1 6 4 8)
• CityList2 (2 5 7 6 8 1 3 4)

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Mutating Permutations

• Changing just one entry in the permutation would
  give an inadmissible solution
• Alternative
• Pick two allele values at random
• Move the second to follow the first, shifting
  the rest along to accommodate
• Note that this preserves most of the order and
  the adjacency information

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Crossover operators for permutations

• Normal crossover operators will often lead to
  inadmissible solutions
• Many specialised operators have been devised
  which focus on combining order or adjacency
  information from the two parents

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Order 1 Crossover

• Idea is to preserve relative order that elements
  occur
• Informal procedure
• 1. Choose an arbitrary part from the first parent
• 2. Copy this part to the first child
• 3. Copy the numbers that are not in the first
  part, to the first child
• starting right from cut point of the copied part,
  
• using the order of the second parent
• and wrapping around at the end
• 4. Analogous for the second child, with parent
  roles reversed

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TSP Example 30 Cities
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Solution i (Distance 941)
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Solution j(Distance 800)
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Solution k(Distance 652)
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Best Solution (Distance 420)
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Overview of Performance
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Hardware Software Optimization

• GA have been particularly successful for finding
  small circuits or code snippets that implement
  common functions
• E.g. Sorting network parallel circuit that
  sorts a fixed number of inputs using compare /
  exchange operators

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Representation

• For design tasks like this, the genotype is not
  just a low-level bit-string, but a high-level
  data structure with meaningful sub-structure
• Software parse tree
• Circuit treat as a program
• gates functions
• wires variables

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Swarm Algorithms, Briefly

• Idea
• Each insect in a swarm is local search process
• At each step, each insect
• Looks around its neighborhood
• Decides which direction looks best
• Communicates what it found out to (some of) the
  other insects
• According to a random coin flip,
• Moves in the direction that looks best locally
• Moves in the best direction it hears about
• Moves in some weighted combination of the above

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Example Particle Swarm Optimization
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Constraint Satisfaction
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Guessing versus Reasoning

• A central feature of all search algorithms is
  making guesses
• Which node to expand?
• Which child to explore?
• We can reduce the amount of guesswork by doing
  more reasoning about the consequences of past
  decisions
• Reasoning about past choices can prune away many
  future choices

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Discrete Constraint Satisfaction Problem

• X is a set of variables x1, x2, , xn
• D is a set of finite domains D1, D2, , Dn
• C is a set of constraints C1, C2, , Cm
• Each constraint restricts the joint values of a
  subset of the variables
• Example 3-Coloring
• Xi countries
• Di Red, Blue, Green
• For each adjacent xi, xj there is a constraint
  Ck(xi,xj) ? (R,G), (R,B), (G,R), (G,B), (B,R),
  (B,G)

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Kinds of Problems

• Find a solution that satisfies all constraints
• Find all solutions
• Find a tightest form for each constraint
• (x1,x2) ? (R,G), (R,B), (G,R), (G,B), (B,R),
  (B,G)
• ?
• (x1,x2) ? (R,G), (R,B)
• Find a solution that minimizes some additional
  objective function

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Solving CSP by Depth-First Search

• Interleave inference and guessing
• At each internal node
• Select unassigned variable
• Select a value in domain
• Backtracking try another value
• At each node
• Propagate Constraints

Where are the guesses?
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4 Queens

• Q4 ? 1,2,3,4
• Q3 ? 1,2,3,4
• Q2 ? 1,2,3,4
• Q1 ? 1,2,3,4

4
3
2
1
1 2 3 4
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Constraint Checking
Takes 5 guesses to determine first guess was wrong
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Forward Checking
When variable is set, immediately remove
inconsistent values from domains of other
variables
Takes 3 guesses to determine first guess was wrong
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Arc Consistency
4
3

• Iterated forward checking

2
1
1 2 3 4
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Arc Consistency
4
3

• Reduce domain of Q1

2
1
1 2 3 4
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Arc Consistency
4
3

• Reduce domain of Q2

2
1
1 2 3 4
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Arc Consistency
4
3

• Reduce domain of Q3

2
1
1 2 3 4
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Arc Consistency
4
3

• Reduce domain of Q4

2
1
1 2 3 4
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Arc Consistency
4
3

• Note that Q32 is determined by (Q2,Q3)

2
1
1 2 3 4
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Arc Consistency
4
3

• Propagating Q32 eliminates all rows in (Q3,Q4)

2
1
1 2 3 4
violated constraint!
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Arc Consistency
4
3

• First choice Q11 shown bad with no further
  guessing!

2
1
1 2 3 4
violated constraint!
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Path Consistency

• Path consistency (3-consistency)
• Iterated check of every triple of variables
• K-consistency
• N-consistency backtrack-free search

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Variable and Value Selection

• Select variable with smallest domain
• Minimize branching factor
• Most likely to propagate, reduce guessing
• Most constrained variable heuristic
• Which values to try first?
• Most likely value for solution
• Forward checking reduces fewest constraints
• Least constrained value heuristic

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CSP Applications

• Scheduling
• NASA Shuttle repair, Hubble telescope
  scheduling, ...
• College basketball scheduling
• Configuration
• 5ESS Switching System gt200 options, highly
  complex interactions
• Reduced configuration time from 2 months ? 2
  hours

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Router Design

• Dynamic wavelength router design
• each channel cannot be repeated in the same
  input port (row constraints)
• each channel cannot be repeated in the same
  output port (column constraints)

(Barry and Humblet 93, Cheung et al. 90, Green
92, Kumar et al. 99)