An Introduction to CP

Transcription

1 An Introduction to CP

2 CP = A technique to solve CSPs and COPs CSP = Constraint Satisfaction Problem COP = Constraint Optimization Problem It. Problema di Soddisfacimento/Ottimizzazione di/con Vincoli A declarative approach Model & then solve (a bit like MIP) Model = variables + constraints Rich set of constraints (much more than linear in-equalities) Is it worth learning?

3 Netherlands railways: 5,500 trains per day In 2009: timetable complete redesign (OR and CP) Less delay, more trains, profit increase: ~\$75M

4 Port of Singapore: > 200 shipping lines, > 600 connected ports Problem: yard location assignment, loading plans For years, the Yard planning system (CP based) assisted the job

5 Rosetta-Philae mission In 2014, first (partially successful) probe landing on a Comet Probe-spacecraft communication pre-scheduled via CP

6 A few reasons for using CP: Rich modeling language Fast prototyping Easy to maintain (modifications are simple) Extensible (new constraints, customizable search...) Very good framework for hybrid approaches Overall: It's a good solution technique! Especially for messy, real-world, problems

7 Before we tackle complex stuff Let's take our first steps...

8 4 available colors, different colors for contiguous regions How would you solve it?

9 4 available colors, different colors for contiguous regions Pick and color a region Pick another region, choose a compatible color Rinse & repeat

10 4 available colors, different colors for contiguous regions Eventually we find something like this

11 Think of that as "Poor's man sudoku" Different numbers (1 to 4) on rows and columns 2 4 How would you solve it?

12 Think of that as "Poor's man sudoku" Different numbers (1 to 4) on rows and columns 2 4 Pick a cell, insert compatible value Rinse & repeat

13 Think of that as "Poor's man sudoku" Different numbers (1 to 4) on rows and columns What now?

14 Think of that as "Poor's man sudoku" Different numbers on rows and columns Clear one of more moves Restart to insert numbers

15 Think of that as "Poor's man sudoku" Different numbers on rows and columns Eventually, we find something like this

16 You see? There is a pattern. We reason on the constraints to narrow down the possible choice We make (and unmake) some choices The core ideas in Constraint Programming are the same! To the point that for many people: CP = constraint reasoning + search

17 You see? There is a pattern. We reason on the constraints to narrow down the possible choice We make (and unmake) some choices The core ideas in Constraint Programming are the same! But I am not many people! I think this formula is much better: CP = model + constraint reasoning + search CP is a declarative approach, remember? So it all starts with a declarative model In the next slides we will focus on each of the three aspects

18 CP = model +...

19 First, we need to define the kind of problem we are interested in: where: is a set of variables is a single variable In principle: any kind of variable! is a set of domains takes values in is a set of constraints

20 Constraint scope (Italian: ambito) A constraint is defined over a subset of of the variables is called the scope of the constraint Tuple A tuple or arity is just a sequence of values Relation Let be a sequence of sets A relation over is a subset of the Cartesian product. I.e.:

21 Constraint A constraint is a relation over the domains of. I.e.: Here's the main idea: A tuple in the Cartesian product = an assignment of the variables A constraint is just a list of feasible assignments A bit abstract, so let's make an example

22 Variables: Domains:, Constraints:

23 A solution for a CSP is an assignment of all the variables that is feasible for all constraints. Formally: where: is the projection of over = sequence of values assigned by to variables in

24 A solution for a CSP is an assignment of all the variables that is feasible for all constraints. Formally: The solutions in our example:,,, A CSP with no solution is called infeasible.

25 Any kind of domain?

26 Any kind of domain? In practice: Integer variables Real variables Set variables! (e.g. ) Graph variables!... In this course: Strong emphasis on finite, integer domains The most studied case

27 Any kind of constraints?

28 Any kind of constraints? With finite domain variables, yes! We can actually list all the feasible assignments This is called extensional representation Very general Possibly inconvenient and inefficient (large domains) We may prefer a symbolic or intensional representation: More compact, more clear, less general

29 Intensional forms are made possible by Constraint libraries Constraint library = collection of constraint types Our first constraint library: Equality: Notation: Semantic: satisfied if is equal to Disequality: Notation: Semantic: satisfied if is not equal to

30 Variables and domains:, for each region Constraints: if region and are contiguous

31 2 4 Variables and domains:, for each cell = row index, = column index Constraints:

32 2 4 Variables and domains:, for each cell = row index, = column index Constraints: is cell is pre-filled with value

33 CP = model + constraint reasoning +...

34 2 4 We would never put a there (not compatible) Can we formalize this deduction? Main idea: reason on the domains

35 2 4

36 2 4 1 The only possible value for is 1

37 Filtering for = removing provably infeasible values from the domains of the variables in the constraint scope Alternative terms: pruning, propagation My convention: Pruning = the act of removing a single value Filtering = the process that guides pruning Propagation = later! Italian: propagation = propagazione

38 Filtering for = removing provably infeasible values from the domains of the variables in the constraint scope For constraints in extensional form: General methods (we will see them later) For constraints in intensional form: Specialized filtering algorithms Those are specified in the constraint library Alternative names: filtering rules, propagators

39 Equality constraint: Rule 1: Rule 2: Examples: Before filtering: After filtering:

40 Disequality constraint: Rule 1: Rule 2: Examples: Before filtering: After filtering:

41 2 4 All original domains are Let's filter all the constraints in order By row and then column

42 1 2 4 Constraint prunes 2 from

43 Constraint prunes 2 from

44 Constraints and prune a lot

45 By filtering all constraints in order, we get this Can we do more?

46 Yep: and are now singletons Hence, our filtering rules for the constraints can be triggered If we do, we can filter more

47 Propagation = the process by which filtering from one constraint may enable filtering from another constraint Propagation is controlled by a propagation algorithm

48 Algorithm: AC1 dirty = true while dirty: dirty = false for : if : dirty = true Where is a procedure that: Given the current variable domains......applies the filtering algorithm of......and then returns the updated domains

49 AC1 always converges to a fix point, which is independent on the constraint processing order. Proof: We will prove the result in two steps: First, we show that AC1 always terminates......when no more pruning can be done Second, we show that the processing order does not matter The proof relies on some properties of

50 The function filter( ) is inflationary, i.e.: True because a filtering algorithm can only remove domains values Caveat: Technically, is inflationary w.r.t. the order Hence, the term may sound a bit misleading Consequence: AC1 always terminates In the worst case, when becomes empty

51 The function is monotone, i.e.: True because removing a value from : can only reduce the number of feasible assignments and hence increase the number of values that can be pruned Consequence: the processing order of Whenever a filtering algorithms prunes something......the other algorithms just become stronger does not matter

52 Fix point: what's the point? To relax!

53 Thanks to the fix point theorem: Fix point: what's the point? To relax! We can focus on filtering and let the propagation algorithm handle the rest

54 Warning: the constraint processing order does not affect the fix point does affect the number of AC1 iterations Some orders may be more efficient Can we determine the optimal order? Interesting (and challenging) research question NP-hard in general

55 If we apply AC1 to our PLS

56 If we apply AC1 to our PLS We get this (we pruned a lot!)......and still we don't have a solution :-(

57 CP = model + constraint reasoning + search

58 In general, filtering and propagation are not enough to solve a CSP We still need to search for a solution In fact solving a CSP is NP-hard No surprise, since the definition is so general... Don't forget that: Most of the interesting problems out there are NP-hard!

59 The simplest search approach in CP is Depth First Search: function : if a solution has been found: return true if the CSP is infeasible: return false for in : if : return true return false

60 The simplest search approach in CP is Depth First Search: If the problem is infeasible: return false If a solution has been found: return true Otherwise, build a set of "decisions": Try to apply a decision If successful return true Othewise, backtrack Un-make the last decision Try with the next one If no decision is successful, return false (the problem is infeasible)

61 The simplest search approach in CP is Depth First Search: function : if a solution has been found: return true if the CSP is infeasible: return false for in : if : return true return false In our pseudo-code, backtracks are handled via recursion. In practice, this may be quite inefficient......we'll discuss this point again much later in the course

62 The simplest search approach in CP is Depth First Search: function : if a solution has been found: return true if the CSP is infeasible: return false for in : if : return true return false Unclear points: How to detect if a problem is infeasible? What are the decisions returned by? What does it mean to "apply a decision"?

63 Q1: How to detect if a problem is infeasible? A problem is infeasible if we have a domain wipeout Domain wipeout = empty domain, because of propagation This is a sufficient condition This is not a necessary condition I.e. if there is wipeout, then the problem is infeasible But infeasibility may hold even if there is not wipeout More on this later!

64 Q2: What are the decisions returned by? They are constraints! For example: We pick the first unbound variable Unbound variable = variable with non-singleton domain We pick the smallest value in Our decisions are and There are many more options, but for now let us stick to this.

65 Q: What does it mean to "apply a decision"? It means: Adding (posting) the constraint to the problem Triggering its filtering algorithm for the first time Remember: on backtrack, the constraint must be removed.

66 First unbound variable = Minimum value = 1 We post, and the propagate

67 First unbound variable = Minimum value = 1 We post, and the propagate

68 And we have a solution! With just one search decision! This result is enabled by the use of constraint reasoning

69 A Modeling Excercise

70 Giani, Piero, Marisa e Luisanna, sono quattro (stagionati) abitanti di un piccolo paesello del centro Italia. I quattro sono legati da distanti (ma indiscutibili) legami di parentela e pertanto, come vuole il buon costume, di tanto in tanto per le occasioni si scambiano regali. I quattro non è che vadano proprio d amore e d accordo e così, un po per scherzo ed un po per dispetto, da qualche anno continuano a regalarsi le stesse chincaglierie Nella fattispecie, le chincaglierie sono: una riproduzione in vetro di una conchiglia tropicale, una vecchia clessidra, una lampada in fibre ottiche (di quelle pelose) ed una statuetta a forma di cavalluccio marino.

71 Siamo al terzo anno in cui le cianfrusaglie devono essere scambiate e finora non ci sono stati incidenti diplomatici. All indomani dell inizio del quarto anno, Gianni deve ancora fare il suo regalo, torturato dal timore essere il primo a scatenare uno scandalo. Il poveretto ripensa con preoccupazione agli scambi a cui ha partecipato: Il primo anno, Gianni aveva in casa la conghiglia, che ha regalato a Marisa. Lo stesso anno, Marisa ha regalato a Gianni la lampada. Il secondo anno, Gianni ha regalato la lampada a Piero ed ha ricevuto in dono da Luisanna la clessidra.

72 Fortunatamente, Gianni è venuto a conoscenza tramite suo nipote di una nuova metodologia che si chiama Programmazione a Vincoli, che pare possa aiutarlo a risolvere i suoi problemi... Si modelli il problema dello scambio dei regali tramite Programmazione a Vincoli e si indichi a quale persona Gianni potrà regalare la clessidra senza scatenare una bagarre.

73 Si tratta di un Partial Latin Square problem un po camuffato. Una possibile soluzione consiste nell introdurre una variabile per ogni persona e per ogni anno (incluso il quarto!):

74 Ogni anno, ogni persona possiede esattamente un oggetto: Nessuno può possedere lo stesso oggetto due volte: Alcuni scambi sono noti:

75 Esiste una sola soluzione feasible: Gianni: conchiglia lampada clessidra statuetta Piero: clessidra statuetta lampada conchiglia Marisa: lampada conchiglia statuetta clessidra Luisanna: statuetta clessidra conchiglia lampada Pertanto, Gianni deve regalare la clessidra a Marisa se vuole evitare lo scandalo