תכנון ובעיות " קשות " בינה מלאכותית אבי רוזנפלד

יש בעיות שכנראה המחשב לא יכול לפתור P NP NP-Complete

איך פותרים את הבעיה ??? אין פתרון פולינומיאלי ! צריכים לבדוק כל אפשרות ! BRUTE FORCE = n! תכנות דינאמי – " רק " n 2 2 n 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 !

רק הבעיות המעניות הם NP Deterministic Polynomial Time: The TM takes at most O(n c ) steps to accept a string of length n Non-deterministic Polynomial Time: The TM takes at most O(n c ) steps on each computation path to accept a string of length n

אז מה עושים ? Branch and Bound ( כעין PRUNING) HILLCLIMING HEURISTICS Mixed Integer Programming

Constraint Satisfaction Problems (CSP) Constraint Optimization Problems (COP) Backtracking search for CSP and COPs Problem Structure and Problem Decomposition Local search for COPs 10 דוגמאות CSP ו COP

Variables WA, NT, Q, NSW, V, SA, T Domains D i = {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)} 11 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 12 Example: Map-Coloring

Binary CSP: each constraint relates two variables Constraint graph: nodes are variables, arcs show constraints 13 Constraint graph General-purpose CSP algorithms use the graph structure to speed up search, e.g., Tasmania is an independent subproblem

Discrete variables – finite domains: n variables, domain size d implies O(d n ) 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., StartJob ≤ StartJob 3 Continuous variables – e.g., start/end times for Hubble Space Telescope observations – linear constraints solvable in polynomial time by linear programming methods 14 Varieties of CSPs

Assignment problems – e.g., who teaches what class? Timetabling problems – e.g., which class (or exam) is offered when and where? Hardware Configuration Spreadsheets Transportation scheduling Factory scheduling Floorplanning Notice that many real-world problems involve real- valued variables 15 Real-world CSPs

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! · d n leaves (bad) 16 Standard search formulation (incremental) b is branching factor, d is size of domain, n is number of variables

Let’s say we had 4 variables, each of which could take one of 4 integer values 17 What This Naïve Search Would Look Like ???? 1??? 2??? 3??? 4??? ?1?? ?2???3???4?? ??1???2???3???4????1???2???3???4 Etc.…terrible branching factor 11??12??13??14??1?1?1?2?1?3?1?4?1??11??21??31??43 At start, all unassigned

18 Backtracking example

19 Backtracking example

20 Backtracking example

21 Backtracking example

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 שיפורים

Minimum remaining values (MRV): choose the variable with the fewest legal values 23 Minimum Remaining Values All are the same here Two equivalent ones here Clear choice

Tie-breaker among MRV variables Degree heuristic: choose the variable with the most constraints on remaining variables 24 Degree Heuristic

Given a variable, choose the least constraining value: – the one that rules out the fewest values in the remaining variables 25 Least Constraining Value

Given a variable, choose the most constraining value: – This is VERY constrained, so let’s do it first… 26 Most Constraining Value

Idea: – Keep track of remaining legal values for unassigned variables – Terminate search when any variable has no legal values 27 Forward checking

Tasmania and mainland are independent subproblems Subproblems identifiable as connected components of constraint graph 28 Problem Structure

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 29 Iterative Algorithms for CSPs

Example: Sudoku  Variables:  Each (open) square  Domains:  {1,2,…,9}  Constraints: 9-way alldiff for each row 9-way alldiff for each column 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, J 1,…, J n. Each job j is composed of a sequence of operations O j 1,..., O j Lj 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

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 34 Summary