Constraint Satisfaction Problems

Transcription

1 Constaint Satisfaction Polems Seach and Look-ahead Alet-Ludwigs-Univesität Feiug Stefan Wölfl, Chistian Becke-Asano, and Benhad Neel Noveme 17, 2014

2 Seach and Look-ahead Enfocing consistency is one way of solving constaint netwoks: Gloally consistent netwoks can easily e solved in polynomial time. Howeve, enfocing gloal consistency is costly in time and space: it not only takes exponential time to compute an equivalent gloally consistent netwok, ut also exponential space to stoe it. Thus, it is usually advisale to only enfoce local consistency (e. g., ac consistency o path consistency), and compute a solution though seach though the emaining possiilities. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 2 / 50

3 State Spaces Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 3 / 50

4 State spaces: Infomally The fundamental astactions fo seach ae state spaces. They ae defined in tems of: states, epesenting a patial solution to a polem (which may o may not e extensile to a full solution) an initial state fom which to seach fo a solution goal states epesenting solutions opeatos that define how a new state can e otained fom a given state Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 5 / 50

5 State spaces: Fomally Definition (state space) A state space is a 4-tuple S = S,s 0,S,O, whee S is a finite set of states, s 0 S is the initial state, S S is the set of goal states, and O is a finite set of opeatos, whee each opeato o O is a patial function on S, i. e. o : S S fo some S S. We say that an opeato o is applicale in state s if o(s) is defined. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 6 / 50

6 Seach Seach is the polem of finding a sequence of opeatos that tansfoms the initial state into a goal state. Definition (solution of a state space) Let S = S,s 0,S,O e a state space, and let o 1,...,o n O e an opeato sequence. Inductively define esult states 0, 1,..., n S {invalid}: 0 := s 0 Fo i {1,...,n}, if o i is applicale in i 1, then i := o i ( i 1 ). Othewise, i := invalid. The opeato sequence is a solution iff n S. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 7 / 50

7 Seach gaphs and seach algoithms State spaces can e depicted as state gaphs: laeled diected gaphs whee states ae vetices and thee is a diected ac fom s to s with lael o iff o(s) = s fo some opeato o. Thee ae many classical algoithms fo finding solutions in state gaphs, e. g. depth-fist seach, eadth-fist seach, iteative deepening seach, o heuistic algoithms like A. These algoithms offe diffeent tade-offs in tems of untime and memoy usage. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 8 / 50

8 State spaces fo constaint netwoks The state spaces fo constaint netwoks usually have two special popeties: The seach gaphs ae tees (i. e., thee is exactly one path fom the initial state to any eachale seach state). All solutions ae at the same level of the tee. Due to these popeties, vaiations of depth-fist seach ae usually the method of choice fo solving constaint netwoks. We will now define state spaces fo constaint netwoks. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 9 / 50

9 Unodeed seach space Definition (unodeed seach space) Let N = V,D,C e a constaint netwok. The unodeed seach space of N is the following state space: states: patial solutions of N (i. e., consistent assignments) initial state: the empty assignment /0 goal states: solutions of N opeatos: fo each v i V and d D i, one opeato o vi =d as follows: o vi =d is applicale in those states s whee v i is not defined and s {(v i,d)} is consistent o vi =d(s) = s {(v i,d)} Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 10 / 50

10 Unodeed seach space: Intuition The unodeed seach space fomalizes the systematic constuction of solutions, y consistently extending patial solutions until a solution is found. Late on, we will conside altenative (non-systematic) seach techniques. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 11 / 50

11 Unodeed seach space: Discussion In pactice, one will only seach fo solutions in suspaces of the complete unodeed seach space: Conside a state s whee v i V has not een assigned a value. If no solution can e eached fom any successo state fo the opeatos o vi =d (d D i ), then no solution can e eached fom s. Thee is no point in tying opeatos o vj =d fo othe vaiales v j v i in this case! Thus, it is sufficient to conside opeatos fo one paticula unassigned vaiale in each seach state. How to decide which vaiale to use is an impotant issue. Hee, we fist conside static vaiale odeings. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 12 / 50

12 Odeed seach spaces Let N = V,D,C e a constaint netwok. Definition (vaiale odeing) A vaiale odeing of N is a pemutation of the vaiale set V. We wite vaiale odeings in sequence notation: v 1,...,v n. Definition (odeed seach space) Let σ = v 1,...,v n e a vaiale odeing of N. The odeed seach space of N along odeing σ is the state space otained fom the unodeed seach space of N y esticting each opeato o vi =d i to states s with s = i 1. In othe wods, in the initial state, only v 1 can e assigned, then only v 2, then only v 3,... Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 13 / 50

13 The impotance of good odeings All odeed seach spaces fo the same constaint netwok contain the same set of solution states. Howeve, the total nume of states can vay damatically etween diffeent odeings. The size of a state space is a (ough) measue fo the hadness of finding a solution, so we ae inteested in small seach spaces. One way of measuing the quality of a state space is y counting the nume of dead ends: the fewe, the ette. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 14 / 50

14 Dead ends Definition (dead end) A dead end of a state space is a state which is not a goal state and in which no opeato is applicale. In an odeed seach space, a dead end is a patial solution that cannot e consistently extended to the next vaiale in the odeing. In the unodeed seach space, a dead end is a patial solution that cannot e consistently extended to any of the emaining vaiales. In oth cases, this patial solution cannot e pat of a solution. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 15 / 50

15 Backtack-fee seach spaces Definition (acktack-fee) A state space is called acktack-fee if it contains no dead ends. A constaint netwok N is called acktack-fee along vaiale odeing σ if the odeed seach space of N along σ is acktack-fee. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 16 / 50

16 Backtack-fee netwoks: Discussion Backtack-fee netwoks ae the ideal case fo seach algoithms. Constaint netwoks ae aely acktack-fee along any odeing in the way they ae specified natually. Howeve, constaint netwoks can e efomulated (eplaced with an equivalent constaint netwok) to educe the nume of dead ends. One way of doing this is y enfocing a local consistency popety like ac consistency o path consistency, which leads to a tighte netwok. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 17 / 50

17 Constaint tightness and dead ends Lemma Let N and N e equivalent constaint netwoks. If N is at least as tight as N, then Poof. the unodeed seach space of N has at most as many dead ends as the unodeed seach space of N, and the odeed seach space of N along any odeing σ has at most as many dead ends as the odeed seach space of N along the same odeing σ. Fo evey dead end of N (in eithe kind of state space), the same assignment is a state in the state space fo N which has at least one dead end as a descendant. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 18 / 50

18 Gloal consistency and dead ends Lemma Let N e a constaint netwok. The following thee statements ae equivalent: The unodeed seach space of N is acktack-fee. The odeed seach space of N is acktack-fee along each odeing σ. N is gloally consistent. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 19 / 50

19 Reducing dead ends futhe Replacing constaint netwoks y tighte, equivalent netwoks is a poweful way of educing dead ends. Howeve, one can go much futhe y also tightening constaints duing seach, fo example y enfocing local consistency fo a given patial instantiation. We will conside such seach algoithms soon. In geneal, thee is a tade-off etween educing the nume of dead ends and the ovehead fo consistency easoning. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 20 / 50

20 Backtacking Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 21 / 50

21 Backtacking Backtacking taveses the seach space of patial instantiations in a depth-fist manne in two phases: fowad phase: vaiales ae selected in sequence; the cuent patial solution is extended y assigning a consistent value to the next vaiale (if possile) ackwad phase: if no consistent instantiation fo the cuent vaiale exists, we etun to the pevious vaiale. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 23 / 50

22 Backtacking: Example Conside the constaint netwok defined y the following coloing polem: v 2, g v 1,, g v 7, v 6, g, y, v 3, v 4, g v 5 Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 24 / 50

23 Backtacking: Example On this example we apply the acktacking algoithm y using the vaiale odeing: v 1,v 7,v 4,v 5,v 6,v 3,v 2, and we otain: v 1 v 7 v 4 v 5 v 6 v 3 v 2 Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 25 / 50

24 Backtacking: Example On this example we apply the acktacking algoithm y using the vaiale odeing: v 1,v 7,v 4,v 5,v 6,v 3,v 2, and we otain: v 1 g v 7 v 4 v 5 v 6 v 3 v 2 Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 25 / 50

25 Backtacking: Example On this example we apply the acktacking algoithm y using the vaiale odeing: v 1,v 7,v 4,v 5,v 6,v 3,v 2, and we otain: v 1 g v 7 v 4 v 5 v 6 v 3 v 2 Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 25 / 50

26 Backtacking: Example On this example we apply the acktacking algoithm y using the vaiale odeing: v 1,v 7,v 4,v 5,v 6,v 3,v 2, and we otain: v 1 g v 7 v 4 v 5 v 6 v 3 v 2 Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 25 / 50

27 Backtacking: Example On this example we apply the acktacking algoithm y using the vaiale odeing: v 1,v 7,v 4,v 5,v 6,v 3,v 2, and we otain: v 1 g v 7 v 4 v 5 v 6 v 3 v 2 Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 25 / 50

28 Backtacking: Example On this example we apply the acktacking algoithm y using the vaiale odeing: v 1,v 7,v 4,v 5,v 6,v 3,v 2, and we otain: v 1 g v 7 v 4 v 5 g v 6 v 3 v 2 Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 25 / 50

29 Backtacking: Example On this example we apply the acktacking algoithm y using the vaiale odeing: v 1,v 7,v 4,v 5,v 6,v 3,v 2, and we otain: v 1 g v 7 v 4 v 5 g v 6 v 3 v 2 Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 25 / 50

30 Backtacking: Example On this example we apply the acktacking algoithm y using the vaiale odeing: v 1,v 7,v 4,v 5,v 6,v 3,v 2, and we otain: v 1 g v 7 v 4 v 5 g g v 6 y y v 3 v 2 Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 25 / 50

31 Backtacking: Example On this example we apply the acktacking algoithm y using the vaiale odeing: v 1,v 7,v 4,v 5,v 6,v 3,v 2, and we otain: v 1 g v 7 v 4 v 5 g g v 6 y y v 3 v 2 Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 25 / 50

32 Backtacking: Example On this example we apply the acktacking algoithm y using the vaiale odeing: v 1,v 7,v 4,v 5,v 6,v 3,v 2, and we otain: v 1 g v 7 v 4 v 5 g g v 6 y y v 3 v 2 Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 25 / 50

33 Backtacking: Example On this example we apply the acktacking algoithm y using the vaiale odeing: v 1,v 7,v 4,v 5,v 6,v 3,v 2, and we otain: v 1 g v 7 v 4 v 5 g g v 6 y y v 3 v 2 Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 25 / 50

34 Backtacking: Example On this example we apply the acktacking algoithm y using the vaiale odeing: v 1,v 7,v 4,v 5,v 6,v 3,v 2, and we otain: v 1 g v 7 v 4 v 5 g g v 6 y y v 3 v 2 Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 25 / 50

35 Backtacking: Example On this example we apply the acktacking algoithm y using the vaiale odeing: v 1,v 7,v 4,v 5,v 6,v 3,v 2, and we otain: v 1 g v 7 v 4 v 5 g g v 6 y y v 3 v 2 Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 25 / 50

36 Backtacking algoithm (ecusive vesion) Backtacking(N, a): Input: a constaint netwok N = V, D, C and a patial solution a of N (initially: the empty instantiation a = {}) Output: a solution of N o inconsistent if a is not locally consistent with N: etun inconsistent if a is defined fo all vaiales in V: etun a select some vaiale v i fo which a is not defined fo each value d fom D i : a := a {(v i,d)} a Backtacking(N,a ) if a is not inconsistent : etun a etun inconsistent Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 26 / 50

37 Backtacking algoithm (ecusive vesion 2) Backtacking(N, a): Input: a constaint netwok N = V, D, C and a patial solution a of N (initially: the empty instantiation a = {}) Output: a solution of N o inconsistent if a is defined fo all vaiales in V: etun a select some vaiale v i fo which a is not defined fo each value d fom D i : a := a {(v i,d)} if a is locally consistent with N: a Backtacking(N,a ) if a is not inconsistent : etun a etun inconsistent Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 27 / 50

38 Banching stategies Enumeation The vaiale v is instantiated in tun to each value in its domain. Fist v = d 1, then v = d 2, etc. Binay choice points The vaiale v is instantiated to some value in its domain. Assuming the value 1 is chosen in ou example, two anches ae geneated and the constaints v = d 1 and v d 1 ae posted, espectively. Domain splitting The domain of the vaiale v is split in two pats. Fo instance, with a domain of size 4: choose fist v = {d 1,d 2 }, then v = {d 3,d 4 } Those ae identical when constaints ae inay. Fo this lectue, we will only conside the enumeation anching stategy. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 28 / 50

39 Look-ahead stategies Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 29 / 50

40 Impovements of Backtacking Backtacking suffes fom thashing: patial solutions that cannot e extended to a full solution may e epocessed seveal times (always leading to a dead end in the seach space) Idea: Impove (pactical) pefomance y pepocessing the seach space undeneath the cuently selected vaiale impoving (in a dynamic way) the seach stategy two schemes (elated to the two phases of acktacking seach), namely look-ahead and look-ack stategies Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 31 / 50

41 Look-ahead and look-ack Look-ahead: invoked when next vaiale o next value is selected. Fo example: Which vaiale should e instantiated next? pefe vaiales that impose tighte constaints on the est of the seach space Which value should e chosen fo the next vaiale? maximize the nume of options fo futue assignments Look-ack: invoked when the acktacking step is pefomed afte eaching a dead end. Fo example: How deep should we acktack? avoid ielevant acktack points (y analyzing easons fo the dead end and jumping ack to the souce of failue) How can we lean fom dead ends? ecod easons fo dead ends as new constaints so that the same inconsistencies can e avoided at late stages of the seach Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 32 / 50

42 Look-ahead and look-ack Look-ahead: invoked when next vaiale o next value is selected. Fo example: Which vaiale should e instantiated next? pefe vaiales that impose tighte constaints on the est of the seach space Which value should e chosen fo the next vaiale? maximize the nume of options fo futue assignments Look-ack: invoked when the acktacking step is pefomed afte eaching a dead end. Fo example: How deep should we acktack? avoid ielevant acktack points (y analyzing easons fo the dead end and jumping ack to the souce of failue) How can we lean fom dead ends? ecod easons fo dead ends as new constaints so that the same inconsistencies can e avoided at late stages of the seach Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 32 / 50

43 Backtacking with look-ahead LookAhead(N, a): Input: a constaint netwok N = V, D, C and a patial solution a of N (initialy: the empty instantiation a = {}) Output: a solution of N o inconsistent SelectValue(v i,d i,a,n): pocedue that selects and deletes a consistent value d D i ; etuns d and a efinement of N; etuns null, if all a {(v i,d)} ae inconsistent if a is defined fo all vaiales in V: etun a select some vaiale v i fo which a is not defined N N, D i D i // (wok on a copy) while D i is non-empty d,n SelectValue(v i,d i,a,n ) if d is not null : a LookAhead(N,a {(v i,d)}) if a is not inconsistent : etun a etun inconsistent Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 33 / 50

44 Ac-consistency ased look-ahead 1 Fowad Checking: popagate the effect of a value-selection to each single non-instantiated vaiales 2 Patial Look-Ahead... 3 Full Look-Ahead... 4 Real Full Look-Ahead: enfoce full ac consistency on the futue vaiales afte each assignment to the cuent vaiale Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 34 / 50

45 SelectValue-FowadChecking SelectValue-FowadChecking(v i,d i,a,n): select and delete d fom D i fo each v j shaing a constaint with v i fo which a is not defined D j D j // (wok on a copy) fo each value d D j if not consistent(a {(v i,d),(v j,d )}) emove d fom D j if any futue D j is empty // (v i x leads to a dead end) etun null D j D j // (popagate all futue D j ) etun d Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 35 / 50

46 SelectValue-RealFullLookAhead SelectValue-RealFullLookAhead(v i,d i,a,n): select and delete d fom D i D j D j (fo all non-assigned v j v i wok on a copy) epeat fo each v j (j i) fo which a is not defined fo each v k (k i,j) fo which a is not defined fo each value d D j if thee is no value d D k such that consistent(a {(v i,d), (v j,d ), (v k,d )}) emove d fom D j until no value was emoved if any futue D j is empty // (v i d leads to a dead end) etun null D j D j // (popagate all futue D j ) etun d Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 36 / 50

47 Look-ahead example (no look-ahead) Example s 1 s 2 s 3 s 4 s 5 s 1 s 2 s 3 s 4 s 5 Initial State Red Blue Geen Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 37 / 50

48 Example: Look-ahead with fowad checking Example s 1 s 2 s 3 s 4 s 5 s 1 s 2 s 3 s 4 s 5 Initial State Red Blue Geen Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 38 / 50

49 Example: Look-ahead with fowad checking Example s 1 s 2 s 3 s 4 s 5 s 1 s 2 s 3 s 4 s 5 Decision Red Blue Geen O Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 38 / 50

50 Example: Look-ahead with fowad checking Example s 1 s 2 s 3 s 4 s 5 s 1 s 2 s 3 s 4 s 5 Popagation Red Blue Geen O X X X Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 38 / 50

51 Example: Look-ahead with fowad checking Example s 1 s 2 s 3 s 4 s 5 Red Blue Geen s 1 O s 2 X O s 3 s 4 s 5 Decision X X Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 38 / 50

52 Example: Look-ahead with fowad checking Example s 1 s 2 s 3 s 4 s 5 Red Blue Geen s 1 O s 2 X O s 3 X s 4 X s 5 X X Popagation Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 38 / 50

53 Example: Look-ahead with fowad checking Example s 1 s 2 s 3 s 4 s 5 Red Blue Geen s 1 O s 2 X O s 3 X O s 4 X s 5 X X Decision Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 38 / 50

54 Example: Look-ahead with fowad checking Example s 1 s 2 s 3 s 4 s 5 Popagation Red Blue Geen s 1 O s 2 X O s 3 X O s 4 X X s 5 X X X Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 38 / 50

55 Example: Look-ahead with fowad checking Example s 1 s 2 s 3 s 4 s 5 Red Blue Geen s 1 O s 2 X O s 3 O X s 4 X s 5 X X Decision Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 38 / 50

56 Example: Look-ahead with fowad checking Example s 1 s 2 s 3 s 4 s 5 Decision Red Blue Geen s 1 O s 2 X O s 3 O X s 4 X O s 5 X X Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 38 / 50

57 Example: Look-ahead with fowad checking Example s 1 s 2 s 3 s 4 s 5 Decision Red Blue Geen s 1 O s 2 X O s 3 O X s 4 X O s 5 X X O Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 38 / 50

58 Example: Real full look-ahead Example s 1 s 2 s 3 s 4 s 5 s 1 s 2 s 3 s 4 s 5 Initial State Red Blue Geen Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 39 / 50

59 Example: Real full look-ahead Example s 1 s 2 s 3 s 4 s 5 s 1 s 2 s 3 s 4 s 5 Decision Red Blue Geen O Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 39 / 50

60 Example: Real full look-ahead Example s 1 s 2 s 3 s 4 s 5 s 1 s 2 s 3 s 4 s 5 Popagation Red Blue Geen O X X X Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 39 / 50

61 Example: Real full look-ahead Example s 1 s 2 s 3 s 4 s 5 Red Blue Geen s 1 O s 2 X O s 3 s 4 s 5 Decision X X Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 39 / 50

62 Example: Real full look-ahead Example s 1 s 2 s 3 s 4 s 5 Red Blue Geen s 1 O s 2 X O s 3 X s 4 X s 5 X X Popagation Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 39 / 50

63 Example: Real full look-ahead Example s 1 s 2 s 3 s 4 s 5 Red Blue Geen s 1 O s 2 X O s 3 X s 4 X s 5 X X O Popagation Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 39 / 50

64 Example: Real full look-ahead Example s 1 s 2 s 3 s 4 s 5 Popagation Red Blue Geen s 1 O s 2 X O s 3 X X s 4 X X s 5 X X O Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 39 / 50

65 Example: Real full look-ahead Example s 1 s 2 s 3 s 4 s 5 Popagation Red Blue Geen s 1 O s 2 X O s 3 O X X s 4 X O X s 5 X X O Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 39 / 50

66 Ac consistency-ased look-ahead 1 Fowad checking: O(e k 2 ), Real full look-ahead (also known as MAC): with AC3-vaiant in O(e k 3 ), whee k is the cadinality of the lagest domain and e is the nume of constaints. Remak Keeping the alance etween puning the seach space and cost of look-ahead Good tadeoffs ae nowadays: Fowad checking Real full look-ahead Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 40 / 50

67 Futhe SelectValue functions Dynamic look-ahead value odeings: estimate likelihood that a non-ejected value leads to a solution. Fo example: MinConflicts (MC): pefe a value that emoves the smallest nume of values fom the domains of futue vaiales MaxDomainSize (MD): pefe a value that ensues the lagest minimum domain sizes of futue vaiales (i.e., calculate n d := min vj D j afte assigning v i d, and n d fo v i d, espectively; if n d > n d, then pefe v i d) Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 41 / 50

68 Choosing a vaiale ode Backtacking and LookAhead leave the choice of vaiale odeing open. Odeing geatly affects pefomance. execises We distinguish Dynamic odeing: In each state, decide independently which vaiale to assign to next. Can e seen as seach in a suspace of the unodeed seach space. Static odeing: A vaiale odeing σ is fixed in advance. Seach is conducted in the odeed seach space along σ. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 42 / 50

69 Dynamic vaiale odeings Common heuistic: Fail-fist Always select a vaiale whose emaining domain has a minimal nume of elements. intuition: few sutees small seach space exteme case: only one value left no seach compae Unit Popagation in DPLL pocedue Should e comined with a constaint popagation technique such as Fowad Checking o Ac Consistency. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 43 / 50

70 Static vaiale odeings Static vaiale odeings... lead to no ovehead duing seach ut ae less flexile than dynamic odeings In pactice, they ae often vey good if chosen popely. Popula choices: Max-cadinality odeing Min-width odeing Cycle cutset odeing Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 44 / 50

71 Static vaiale odeings: Max-cadinality odeing Max-cadinality odeing 1 Stat with an aitay vaiale. 2 Repeatedly add a vaiale such that the nume of constaints whose scope is a suset of the set of added vaiales is maximal. Beak ties aitaily. fo the othe two odeing stategies, we fist need to lay some foundations Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 45 / 50

72 Odeed gaphs Definition (odeed gaph) Let G = V,E e a gaph. An odeed gaph fo G is a tuple V,E,σ, whee σ is an odeing (pemutation) of the vetices in V. As usual, we use sequence notation fo the odeing: σ = v 1,...,v n. We wite v v if v pecedes v in σ. The paents of v V in the odeed gaph ae the neighos that pecede it: {u V u v,{u,v} E}. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 46 / 50

73 Width of a gaph Definition (width) The width of a vetex v of an odeed gaph is the nume of paents of v. The width of an odeed gaph is the maximal width of its vetices. The width of a gaph G is the minimal width of all odeed gaphs fo G. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 47 / 50

74 Gaphs of width 1 Theoem A gaph with at least one edge has width 1 iff it is a foest (i.e., iff it contains no cycles). Poof. A gaph with at least one edge has at least width 1. ( ): If a gaph has a cycle consisting of vetices C, then in any odeing σ, one of the vetices in C will appea last. This vetex will have width at least 2. Thus, the width of the odeing cannot e 1. ( ): Conside a gaph V,E with no cycles. In evey connected component, pick an aitay vetex; these ae called oot nodes. Constuct odeed gaph V,E,σ y putting oot nodes fist in σ, then nodes with distance 1 fom a oot node, then distance 2, 3, etc. This odeed gaph has width 1. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 48 / 50

75 Gaphs of width 1 Theoem A gaph with at least one edge has width 1 iff it is a foest (i.e., iff it contains no cycles). Poof. A gaph with at least one edge has at least width 1. ( ): If a gaph has a cycle consisting of vetices C, then in any odeing σ, one of the vetices in C will appea last. This vetex will have width at least 2. Thus, the width of the odeing cannot e 1. ( ): Conside a gaph V,E with no cycles. In evey connected component, pick an aitay vetex; these ae called oot nodes. Constuct odeed gaph V,E,σ y putting oot nodes fist in σ, then nodes with distance 1 fom a oot node, then distance 2, 3, etc. This odeed gaph has width 1. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 48 / 50

76 Gaphs of width 1 Theoem A gaph with at least one edge has width 1 iff it is a foest (i.e., iff it contains no cycles). Poof. A gaph with at least one edge has at least width 1. ( ): If a gaph has a cycle consisting of vetices C, then in any odeing σ, one of the vetices in C will appea last. This vetex will have width at least 2. Thus, the width of the odeing cannot e 1. ( ): Conside a gaph V,E with no cycles. In evey connected component, pick an aitay vetex; these ae called oot nodes. Constuct odeed gaph V,E,σ y putting oot nodes fist in σ, then nodes with distance 1 fom a oot node, then distance 2, 3, etc. This odeed gaph has width 1. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 48 / 50

77 Significance of width To find solutions to constaint netwoks, we ae inteested in the width of the pimal constaint gaph. The width of a gaph is a (ough) difficulty measue. Fo width 1, we can make this moe pecise (next slide). In geneal, thee is a povale elationship etween solution effot and a closely elated measue called induced width. The odeing that leads to an odeed gaph of minimal width is usually a good static vaiale odeing. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 49 / 50

78 Constaint gaphs with width 1 Theoem Let N e a constaint netwok whose pimal constaint gaph has width 1. Then N can e solved in polynomial time. Note: Such a constaint netwok must e inay, as constaints of highe aity 3 induce cycles in the pimal constaint gaph. Lemma Let N e an ac-consistent (nomalized) constaint netwok whose pimal constaint gaph has width 1, and whee all vaiale domains ae non-empty. Then N is acktack-fee along any odeing with width 1. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 50 / 50

79 Constaint gaphs with width 1 Theoem Let N e a constaint netwok whose pimal constaint gaph has width 1. Then N can e solved in polynomial time. Note: Such a constaint netwok must e inay, as constaints of highe aity 3 induce cycles in the pimal constaint gaph. Lemma Let N e an ac-consistent (nomalized) constaint netwok whose pimal constaint gaph has width 1, and whee all vaiale domains ae non-empty. Then N is acktack-fee along any odeing with width 1. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 50 / 50

80 Constaint gaphs with width 1 (ctd.) Poof of the lemma. Let N e such a constaint netwok, and let σ = v 1,...,v n e a width-1 odeing fo N. We must show that all patial solutions of the fom {v 1 d 1,...,v i d i } fo 0 i < n can e consistently extended to vaiale v i+1. Since σ has width 1, the width of v i+1 is 0 o 1. v i+1 has width 0: Thee is no constaint etween v i+1 and any assigned vaiale, so any value in the (non-empty) domain of v i+1 is a consistent extension. v i+1 has width 1: Thee is exactly one vaiale v j {v 1,...,v i } with a constaint etween v j and v i+1. Fo evey choice (v j d j ), thee must e a consistent choice (v i+1 d i+1 ) ecause of ac consistency. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 51 / 50

81 Constaint gaphs with width 1 (ctd.) Poof of the lemma. Let N e such a constaint netwok, and let σ = v 1,...,v n e a width-1 odeing fo N. We must show that all patial solutions of the fom {v 1 d 1,...,v i d i } fo 0 i < n can e consistently extended to vaiale v i+1. Since σ has width 1, the width of v i+1 is 0 o 1. v i+1 has width 0: Thee is no constaint etween v i+1 and any assigned vaiale, so any value in the (non-empty) domain of v i+1 is a consistent extension. v i+1 has width 1: Thee is exactly one vaiale v j {v 1,...,v i } with a constaint etween v j and v i+1. Fo evey choice (v j d j ), thee must e a consistent choice (v i+1 d i+1 ) ecause of ac consistency. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 51 / 50

82 Constaint gaphs with width 1 (ctd.) Poof of the lemma. Let N e such a constaint netwok, and let σ = v 1,...,v n e a width-1 odeing fo N. We must show that all patial solutions of the fom {v 1 d 1,...,v i d i } fo 0 i < n can e consistently extended to vaiale v i+1. Since σ has width 1, the width of v i+1 is 0 o 1. v i+1 has width 0: Thee is no constaint etween v i+1 and any assigned vaiale, so any value in the (non-empty) domain of v i+1 is a consistent extension. v i+1 has width 1: Thee is exactly one vaiale v j {v 1,...,v i } with a constaint etween v j and v i+1. Fo evey choice (v j d j ), thee must e a consistent choice (v i+1 d i+1 ) ecause of ac consistency. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 51 / 50

83 Constaint gaphs with width 1 (ctd.) Poof of the lemma. Let N e such a constaint netwok, and let σ = v 1,...,v n e a width-1 odeing fo N. We must show that all patial solutions of the fom {v 1 d 1,...,v i d i } fo 0 i < n can e consistently extended to vaiale v i+1. Since σ has width 1, the width of v i+1 is 0 o 1. v i+1 has width 0: Thee is no constaint etween v i+1 and any assigned vaiale, so any value in the (non-empty) domain of v i+1 is a consistent extension. v i+1 has width 1: Thee is exactly one vaiale v j {v 1,...,v i } with a constaint etween v j and v i+1. Fo evey choice (v j d j ), thee must e a consistent choice (v i+1 d i+1 ) ecause of ac consistency. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 51 / 50

84 Constaint gaphs with width 1 (ctd.) Poof of the theoem. We can enfoce ac consistency and compute a width 1 odeing in polynomial time. If the esulting netwok has any empty vaiale domains, it is tivially unsolvale. Othewise, y the lemma, it can e solved in polynomial time y the Backtacking pocedue. Remak: Enfocing full ac consistency is actually not necessay; a limited fom of consistency is sufficient. (We do not discuss this futhe.) Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 52 / 50

85 Static vaiale odeings: Min-width odeing Min-width odeing Select a vaiale odeing such that the esulting odeed constaint gaph has minimal width among all choices. Remak: Can e computed efficiently y a geedy algoithm: 1 Choose a vetex v with minimal degee and emove it fom the gaph. 2 Recusively compute an odeing fo the emaining gaph, and place v afte all othe vetices. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 53 / 50

86 Static vaiale odeings: Cycle cutset odeing Definition (cycle cutset) Let G = V,E e a gaph. A cycle cutset fo G is a vetex set V V such that the sugaph induced y V \ V has no cycles. Cycle cutset odeing 1 Compute a (pefealy small) cycle cutset V. 2 Fist ode all vaiales in V (using any odeing stategy). 3 Then ode the emaining vaiales, using a width-1 odeing fo the sunetwok whee the vaiales in V ae emoved. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 54 / 50

87 Cycle cutsets: Remaks If the netwok is inay and the seach algoithm enfoces ac consistency afte assigning to the cutset vaiales, no futhe seach is needed at this point. untime O(k V p( N )) fo some polynomial p Howeve, finding minimum cycle cutsets is NP-had. Even finding appoximate solutions is povaly had. Howeve, in pactice good cutsets can usually e found. Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 55 / 50

88 Liteatue Rina Dechte. Constaint Pocessing, Chaptes 4 and 5, Mogan Kaufmann, 2003 Noveme 17, 2014 Wölfl, Neel and Becke-Asano Constaint Satisfaction Polems 56 / 50