Transcript פתרון על ידי בעיות חיפוש

‫תכנון ובעיות "קשות"‬
‫בינה מלאכותית‬
‫אבי רוזנפלד‬
‫יש בעיות שכנראה המחשב לא יכול לפתור‬
‫‪NP-Complete‬‬
‫‪P‬‬
‫‪NP‬‬
‫איך פותרים את הבעיה???‬
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‫אין פתרון פולינומיאלי!‬
‫צריכים לבדוק כל אפשרות!‬
‫‪n! = BRUTE FORCE‬‬
‫תכנות דינאמי– "רק" ‪n22n‬‬
‫כנראה שאין פתרון "סביר" ‪ -‬פולינומיאלי‬
NP-Complete ‫דוגמאות‬
To prove a problem is NP-Complete show a •
polynomial time reduction from 3-SAT
Other NP-Complete Problems: •
PARTITION –
SUBSET-SUM –
CLIQUE –
HAMILTONIAN PATH (TSP) –
GRAPH COLORING –
MINESWEEPER (and many more) –
‫רוב בעיות של תכנון ואופטימיזציה הם פה!‬
NP ‫רק הבעיות המעניות הם‬
• Deterministic Polynomial Time: The TM takes
at most O(nc) steps to accept a string of length
n
• Non-deterministic Polynomial Time: The TM
takes at most O(nc) steps on each computation
path to accept a string of length n
?‫אז מה עושים‬
)PRUNING ‫ (כעין‬Branch and Bound
HILLCLIMING
HEURISTICS
Mixed Integer Programming
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COP‫ ו‬CSP ‫דוגמאות‬
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Constraint Satisfaction Problems (CSP)
Constraint Optimization Problems (COP)
Backtracking search for CSP and COPs
Problem Structure and Problem
Decomposition
• Local search for COPs
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Example: Map-Coloring
• Variables WA, NT, Q, NSW, V, SA, T
• Domains Di = {red, green, blue}
• Constraints: adjacent regions must have different colors
e.g., WA ≠ NT (if the language allows stating this so succinctly), or
(WA, NT) in {(red,green), (red,blue), (green,red), (green,blue),
(blue,red), (blue,green)}
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Example: Map-Coloring
• This is a solution: complete and consistent assignments (i.e.,
all variables assigned, all constraints satisfied):
e.g., WA = red, NT = green, Q = red, NSW = green, V = red,
SA = blue, T = green
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Constraint graph
• Binary CSP: each constraint relates two variables
• Constraint graph: nodes are variables, arcs show constraints
General-purpose CSP algorithms use the graph structure to speed up
search, e.g., Tasmania is an independent subproblem
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Varieties of CSPs
• Discrete variables
– finite domains:
• n variables, domain size d implies O(dn) complete assignments
• e.g., Boolean CSPs, including Boolean satisfiability (NP-complete)
– infinite domains:
• integers, strings, etc.
• e.g., job scheduling, variables are start/end days for each job
• need a constraint language, e.g., StartJob1 + 5 ≤ StartJob3
• Continuous variables
– e.g., start/end times for Hubble Space Telescope observations
– linear constraints solvable in polynomial time by linear
programming methods
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Real-world CSPs
• Assignment problems
– e.g., who teaches what class?
• Timetabling problems
– e.g., which class (or exam) is offered when and where?
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Hardware Configuration
Spreadsheets
Transportation scheduling
Factory scheduling
Floorplanning
• Notice that many real-world problems involve realvalued variables
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Standard search formulation (incremental)
Let’s start with the straightforward (dumb) approach, then fix it.
States are defined by the values assigned so far.
• Initial state: the empty assignment { }
• Successor function: assign a value to an unassigned variable that
does not conflict with current assignment
 fail if no legal assignments
• Goal test: the current assignment is complete
1. This is the same for all CSPs (good)
2. Every solution appears at depth n with n variables
 use depth-first search
3. Path is irrelevant, so can also use complete-state formulation
4. b = (n - l )d at depth l, hence n! · dn leaves (bad)
5.
b is branching factor, d is size of domain, n is number of variables
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What This Naïve Search Would Look
Like
• Let’s say we had 4 variables, each of which
could take one of 4 integer values
????
At start, all unassigned
4???
1??? 2??? 3???
?1?? ?2?? ?3?? ?4??
11??
12??
13??
14??
1?1?
1?2?
1?3?
??1? ??2? ??3? ??4?
1?4?
1??1
1??2
???1 ???2 ???3 ???4
1??3
1??43
Etc.…terrible
branching factor
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Backtracking example
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Backtracking example
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Backtracking example
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Backtracking example
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‫שיפורים‬
‫• יש כמה דרכים איך לשפר את התהליך‪...‬‬
‫– איזה ענף בודקים ראשון‬
‫– איזה בודקים בפעם השנייה‪ ,‬וכו'‬
‫– האם אפשר לעצור את התהליך באמצע?‬
‫– האם אפשר לנצל מידע על הבעיה?‬
‫– האם אפשר לגזום העץ?‬
‫‪22‬‬
Minimum Remaining Values
• Minimum remaining values (MRV):
choose the variable with the fewest legal values
All are
the same
here
Two
equivalent
ones here
Clear
choice
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Degree Heuristic
• Tie-breaker among MRV
variables
• Degree heuristic: choose the
variable with the most
constraints on remaining
variables
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Least Constraining Value
• Given a variable, choose the least constraining
value:
– the one that rules out the fewest values in the remaining
variables
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Most Constraining Value
• Given a variable, choose the most constraining
value:
– This is VERY constrained, so let’s do it first…
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Forward checking
• Idea:
– Keep track of remaining legal values for unassigned
variables
– Terminate search when any variable has no legal values
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Problem Structure
• Tasmania and mainland are independent
subproblems
• Subproblems identifiable as connected
components of constraint graph
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Iterative Algorithms for CSPs
• Hill-climbing, simulated annealing typically work with
“complete” states, i.e., all variables assigned
• To apply to CSPs:
– use complete states, but with unsatisfied constraints
– operators reassign variable values
• Variable selection: randomly select any conflicted
variable
• Value selection by min-conflicts heuristic:
– choose value that violates the fewest constraints
– i.e., hill-climb with h(n) = total number of violated
constraints
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Example: Sudoku
Variables: 
Each (open) square 
Domains: 
{1,2,…,9} 
Constraints: 
9-way alldiff for each column
9-way alldiff for each row
9-way alldiff for each region
Very Important CSPs: Scheduling
• Many industries. Many multi-million $ decisions. Used
extensively for space mission planning. Military uses.
• People really care about improving scheduling algorithms!
Problems with phenomenally huge state spaces. But for which
solutions are needed very quickly
• Many kinds of scheduling problems e.g.:
– Job shop: Discrete time; weird ordering of operations
possible; set of separate jobs.
– Batch shop: Discrete or continuous time; restricted
operation of ordering; grouping is important.
Scheduling
• A set of N jobs, J1,…, Jn.
• Each job j is composed of a sequence of
operations Oj1,..., OjLj
• Each operation may use resource R, and has a
specific duration in time.
• A resource must be used by a single operation
at a time.
• All jobs must be completed by a due time.
• Problem: assign a start time to each job.
‫יש "מעבר פאזה"‬
‫‪Cheeseman 1993‬‬
Summary
• CSPs are a special kind of search problem:
– states defined by values of a fixed set of variables
– goal test defined by constraints on variable values
• Backtracking = depth-first search with one variable assigned per
node
• Variable ordering (MRV, degree heuristic) and value selection
(least constraining value) heuristics help significantly
• Forward checking prevents assignments that guarantee later
failure
• Constraint propagation (e.g., arc consistency) does additional
work to constrain values and detect inconsistencies
• The CSP representation allows analysis of problem structure
• Tree-structured CSPs can be solved in linear time
• Iterative min-conflicts is usually effective in practice
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