Distributed Constraints

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1 Distributed Constraints José M Vidal Department of Computer Science and Engineering, University of South Carolina January 15, 2010 Abstract Algorithms for solving distributed constraint problems in multiagent systems. Chapter 2.

2 Centralized Outline

3 Centralized Graph Coloring

4 Centralized Graph Coloring

5 Centralized Problem (CSP) Given variables x 1,x 2,...x n with domains D 1,D 2,...D n and a set of boolean constraints P of the form pk(x k1,x k2,...,x kj ) {0,1}, find assignments for all the variables such that no constraints are violated.

6 Centralized Depth First Search for the CSP DEPTH-FIRST-SEARCH-CSP(i, g) 1 if i > n 2 then return g 3 for v D i 4 do if setting x i v does not violate any constraint in P given g 5 then g DEPTH-FIRST-SEARCH-CSP(i + 1,g + {x i v}) 6 if g 7 then return g 8 9 return

7 Distributed Outline

8 Distributed Definition (Distributed Problem (DCSP)) Give each agent one of the variables in a CSP. Agents are responsible for finding a value for their variable and can find out the values of their neighbors via communication

9 Filtering Algorithm Outline

10 Filtering Algorithm REVISE(x i,x j ) 1 old-domain D i 2 for v i D i 3 do if there is no v j D j consistent with v i 4 then D i D i v i 5 if old-domain D i 6 then k {neighbors of i} k.handle-new-domain(i,d i )

11 Filtering Algorithm x 1 x 2 x 3 x 1 x 2 x 3 x 1 x 2 x 3 x 1 x 2 x 3

12 Filtering Algorithm x 1 x 2 x 3 REVISE(x 1,x 3 ) x 1 x 2 x 3 x 1 x 2 x 3 x 1 x 2 x 3

13 Filtering Algorithm x 1 x 2 x 3 REVISE(x 1,x 3 ) x 1 x 2 x 3 x 1 x 2 x 3 x 1 x 2 x 3

14 Filtering Algorithm x 1 x 2 x 3 REVISE(x 1,x 3 ) Communicate x 1 x 2 x 3 x 1 x 2 x 3 x 1 x 2 x 3

15 Filtering Algorithm x 1 x 2 x 3 REVISE(x 1,x 3 ) x 1 x 2 x 3 x 1 x 2 x 3 REVISE(x 2,x 3 ) x 1 x 2 x 3

16 Filtering Algorithm x 1 x 2 x 3 REVISE(x 1,x 3 ) x 1 x 2 x 3 x 1 x 2 x 3 REVISE(x 2,x 3 ) x 1 x 2 x 3

17 Filtering Algorithm x 1 x 2 x 3

18 Filtering Algorithm x 1 x 2 x 3 Filtering fails to detect no-solution.

19 Hyper-Resolution Based Consistency Algorithm Outline

20 Hyper-Resolution Based Consistency Algorithm Definition (k-consistency) Given any instantiation of any k 1 variables that satisfy all constraints it is possible to find an instantiation of any k th variable such that all k variable values satisfy all constraints.

21 Hyper-Resolution Based Consistency Algorithm Definition (Strongly k-consistent) A problem is strongly k-consistent if it is j-consistent for all j k.

22 Hyper-Resolution Based Consistency Algorithm Definition (Hyper-Resolution Rule) A 1 A 2 A m (A 1 A 11 ) (A 2 A 21 ) : (A m A m1 ) (A 11 A 21 A m1 ).

23 Hyper-Resolution Based Consistency Algorithm x 1 x 2 x 3 x 1 = x 1 = x 2 = x 2 = x 3 = x 3 = (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) x 1 x 2 x 3

24 Hyper-Resolution Based Consistency Algorithm x 1 x 2 x 3 x 1 = x 1 = x 2 = x 2 = x 3 = x 3 = (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 2 = x 3 = ) x 1 x 2 x 1 x 3 x 1 = x 1 = (x 1 = x 2 = ) (x 1 = x 3 = ) (x 2 = x 3 = )

25 Hyper-Resolution Based Consistency Algorithm x 1 x 2 x 3 x 1 = x 1 = x 2 = x 2 = x 3 = x 3 = (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 2 = x 3 = ) (x 2 = x 3 = ) (x 2 = x 3 = ) x 1 x 1 x 2 x 3 Sends (x 2 = x 3 = ) to x 2 and x 3.

26 Hyper-Resolution Based Consistency Algorithm x 1 x 2 x 3 x 1 = x 1 = x 2 = x 2 = x 3 = x 3 = (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 2 = x 3 = ) (x 2 = x 3 = ) (x 2 = x 3 = ) (x 3 = ) x 2 x 2 x 1 x 3 x 2 = x 2 = (x 2 = x 3 = ) (x 2 = x 3 = ) (x 3 = )

27 Hyper-Resolution Based Consistency Algorithm x 1 x 2 x 3 x 1 = x 1 = x 2 = x 2 = x 3 = x 3 = (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 2 = x 3 = ) (x 2 = x 3 = ) (x 2 = x 3 = ) (x 2 = x 3 = ) (x 3 = ) x 1 x 2 x 1 x 3 x 1 = x 1 = (x 1 = x 2 = ) (x 1 = x 3 = ) (x 2 = x 3 = )

28 Hyper-Resolution Based Consistency Algorithm x 1 x 2 x 3 x 1 = x 1 = x 2 = x 2 = x 3 = x 3 = (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 2 = x 3 = ) (x 2 = x 3 = ) (x 2 = x 3 = ) (x 2 = x 3 = ) (x 3 = ) (x 2 = x 3 = ) (x 2 = x 3 = ) x 1 x 1 x 2 x 3 Sends (x 2 = x 3 = ) to x 2 and x 3.

29 Hyper-Resolution Based Consistency Algorithm x 1 x 2 x 3 x 1 = x 1 = x 2 = x 2 = x 3 = x 3 = (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 2 = x 3 = ) (x 2 = x 3 = ) (x 2 = x 3 = ) (x 2 = x 3 = ) (x 3 = ) (x 2 = x 3 = ) (x 2 = x 3 = ) (x 3 = ) x 2 x 2 x 1 x 3 x 2 = x 2 = (x 2 = x 3 = ) (x 2 = x 3 = ) (x 3 = )

30 Hyper-Resolution Based Consistency Algorithm x 1 x 2 x 3 x 1 = x 1 = x 2 = x 2 = x 3 = x 3 = (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 2 = x 3 = ) (x 2 = x 3 = ) (x 2 = x 3 = ) (x 2 = x 3 = ) (x 3 = ) (x 2 = x 3 = ) (x 2 = x 3 = ) (x 3 = ) (x 3 = ) (x 3 = ) x 1 x 2 x 2 x 3 Sends (x 3 = ) and (x 3 = ) to x 3.

31 Hyper-Resolution Based Consistency Algorithm x 1 x 2 x 3 x 1 = x 1 = x 2 = x 2 = x 3 = x 3 = (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 1 = x 2 = ) (x 2 = x 3 = ) (x 2 = x 3 = ) (x 2 = x 3 = ) (x 2 = x 3 = ) (x 3 = ) (x 2 = x 3 = ) (x 2 = x 3 = ) (x 3 = ) (x 3 = ) (x 3 = ) x 3 x 2 x 1 x 3 x 3 = x 3 = (x 3 = ) (x 3 = ) Contradiction

32 Asynchronous Backtracking Outline

33 Asynchronous Backtracking Backtracking

34 Asynchronous Backtracking Backtracking

35 Asynchronous Backtracking Backtracking

36 Asynchronous Backtracking Backtracking

37 Asynchronous Backtracking Backtracking

38 Asynchronous Backtracking Backtracking?

39 Asynchronous Backtracking Backtracking

40 Asynchronous Backtracking Backtracking

41 Asynchronous Backtracking Agent Variables priority: the agent s fixed priority number. All agents are ordered. local-view: current values of other agents variables. current-value: current value of agent s variable. neighbors: initially, the set of agents with whom agent shares a constraint.

42 Asynchronous Backtracking Remote Calls HANDLE-OK?(j,x j ) This message asks the receiver if that assignment does not violate any of his constraints. HANDLE-NOGOOD(j, nogood) which means that j is reporting that it can t find a value for his variable because of nogood. HANDLE-ADD-NEIGHBOR(j) which requests the agent to add some other agent j as its neighbor.

43 Asynchronous Backtracking HANDLE-OK?(j,x j ) 1 local-view local-view +(j,x j ) 2 CHECK-LOCAL-VIEW()

44 Asynchronous Backtracking CHECK-LOCAL-VIEW() 1 if local-view and x i are not consistent 2 then if no value in D i is consistent with local-view 3 then BACKTRACK() 4 else select d D i consistent with local-view 5 x i d 6 k neighbors k.handle-ok?(i,x i )

45 Asynchronous Backtracking HANDLE-NOGOOD(j, nogood) 1 record nogood as a new constraint 2 for (k,x k ) nogood where k neighbors 3 do k.handle-add-neighbor(i) 4 neighbors neighbors +k 5 local-view local-view +(k,x k ) 6 old-value x i 7 CHECK-LOCAL-VIEW() 8 if old-value x i 9 then j.handle-ok?(i,x i )

46 Asynchronous Backtracking BACKTRACK() 1 nogoods {V V = inconsistent subset of local-view using hyper-res 2 if an empty set is an element of nogoods 3 then broadcast that there is no solution 4 terminate this algorithm 5 for V nogoods 6 do select (j,x j ) where j has lowest priority in V 7 j.handle-nogood(i, V) 8 local-view local-view (j,x j ) 9 CHECK-LOCAL-VIEW()

47 Asynchronous Backtracking Example x 1 x 2 x 3

48 Asynchronous Backtracking Example x 1 x 2 x 3 x 2 x 3

49 Asynchronous Backtracking Example x 1 ok?(x 2, ) ok?(x 3, ) x 2 x 3 x 2 x 3

50 Asynchronous Backtracking Example x 1 local-view = (x 2, ),(x 3, ) ok?(x 2, ) ok?(x 3, ) x 2 x 3 x 2 x 3

51 Asynchronous Backtracking Example x 1 local-view = (x 2, ),(x 3, ) nogood(x 2 = x 3 = ) x 2 x 3 x 2 x 3

52 Asynchronous Backtracking Example x 1 local-view = (x 2, ),(x 3, ) nogood(x 2 = x 3 = ) Added link x 2 x 3 x 2 x 3

53 Asynchronous Backtracking Example x 1 local-view = (x 2, ),(x 3, ) Added link x 2 x 3 x 2 ok?(x 2, ) x 3 nogood(x 2 = x 3 = ) local-view = (x 2, )

54 Asynchronous Backtracking Example x 1 local-view = (x 2, ),(x 3, ) nogood(x 2 = x 3 = ) Added link x 2 x 3 x 2 nogood(x 2, ) x 3 local-view = (x 2, )

55 Asynchronous Backtracking Theorem (ABT is Complete) The ABT algorithm always finds a solution if one exists and terminates with the appropriate message if there is no solution. Proof. By induction. First show that the agent with the highest priority never enters an infinite loop. Then show that given that all the agents with lower priority that k never fall into an infinite loop then k will not fall into an infinite loop.

56 Asynchronous Weak-Commitment Search Outline

57 Asynchronous Weak-Commitment Search Asynchronous Weak-Commitment (AWC) Use dynamic priorities. Change ok? messages to include agent s current priority. Use min-conflict heuristic.

58 Asynchronous Weak-Commitment Search CHECK-LOCAL-VIEW 1 if x i is consistent with local-view 2 then return 3 if no value in D i is consistent with local-view 4 then BACKTRACK() 5 else select d D i consistent with local-view and which minimizes constraint violations with lower priority agents. 6 x i d 7 k neighbors k.handle-ok?(i,x i,priority)

59 Asynchronous Weak-Commitment Search BACKTRACK 1 generate a nogood V 2 if V is empty nogood 3 then broadcast that there is no solution 4 terminate this algorithm 5 if V is a new nogood 6 then (k,xk ) V k.handle-nogood(i,j,priority) 7 priority 1 + max{neighbors priorities} 8 select d D i consistent with local-view and which minimizes constraint violations with lower priority agents. 9 x i d 10 k neighbors k.handle-ok?(i,x i,priority)

60 Asynchronous Weak-Commitment Search Example priority = 0 x 1 x 2 x 3 priority = 0 priority = 0

61 Asynchronous Weak-Commitment Search Example priority = 0 x 1 x 2 x 3 x 2 x 3 priority = 0 priority = 0

62 Asynchronous Weak-Commitment Search Example priority = 0 x 1 ok?(x 2,,0) ok?(x 3,,0) x 2 x 3 x 2 x 3 priority = 0 priority = 0

63 Asynchronous Weak-Commitment Search Example priority = 0 x 1 local-view = (x 2, ),(x 3, ) ok?(x 2,,0) ok?(x 3,,0) x 2 x 3 x 2 x 3 priority = 0 priority = 0

64 Asynchronous Weak-Commitment Search Example priority = 1 x 1 local-view = (x 2, ),(x 3, ) nogood(x 2 = x 3 = ),1 x 2 x 3 x 2 x 3 priority = 0 priority = 0

65 Asynchronous Weak-Commitment Search Example priority = 1 x 1 local-view = (x 2, ),(x 3, ) nogood(x 2 = x 3 = ),1 Added link x 2 x 3 x 2 priority = 0 priority = 0

66 Asynchronous Weak-Commitment Search Example priority = 1 x 1 local-view = (x 2, ),(x 3, ) ok?(x 2,,0) nogood(x 2 = x 3 = ),1 x 2 Added link x 3 local-view = (x 2, ) priority = 0 priority = 0

67 Asynchronous Weak-Commitment Search Example priority = 1 x 1 local-view = (x 2, ),(x 3, ) nogood(x 2 = x 3 = ),1 Added link x 2 x 3 x 2 priority = 0 nogood(x 2, ) priority = 2 local-view = (x 2, )

68 Asynchronous Weak-Commitment Search Theorem (AWC is complete) The AWC algorithm always finds a solution if one exists and terminates with the appropriate message if there is no solution. Proof. The priority values are changed if and only if a new nogood is found. Since the number of possible nogoods is finite the priority values cannot be changed indefinitely. When the priority values are stable AWC becomes ABT, which is complete.

69 Distributed Breakout Outline

70 Distributed Breakout Hill Climbing

71 Distributed Breakout Hill Climbing?

72 Distributed Breakout Hill Climbing

73 Distributed Breakout Hill Climbing?

74 Distributed Breakout Hill Climbing

75 Distributed Breakout Hill Climbing?

76 Distributed Breakout Hill Climbing

77 Distributed Breakout Definition (Quasi-local-minimum) An agent x i is in a quasi-local-minimum if it is violating some constraint and neither it nor any of its neighbors can make a change that results in lower cost for all.

78 Distributed Breakout Remote Procedure Calls HANDLE-OK?(i,x i ) where i is the agent and x i is its current value, HANDLE-IMPROVE(i, improve, eval) where improve is the maximum i could gain by changing to some other color and eval is its current cost.

79 Distributed Breakout HANDLE-OK?(j,x j ) 1 received-ok[j] TRUE 2 agent-view agent-view +(j,x j ) 3 if k neighbors received-ok[k] = TRUE 4 then SEND-IMPROVE() 5 k neighbors received-ok[k] FALSE

80 Distributed Breakout SEND-IMPROVE() 1 cost evaluation of x i given current weights and values. 2 my-improve possible maximal improvement 3 new-value value that gives maximal improvement 4 k neighbors k.handle-improve(i,my-improve,cost)

81 Distributed Breakout HANDLE-IMPROVE(j, improve, eval) 1 received-improve[j] improve 2 if k neighbors received-improve[k] NONE 3 then SEND-OK 4 agent-view 5 k neighbors received-improve[k] NONE

82 Distributed Breakout SEND-OK() 1 if k neighbors my-improve received-improve[k] 2 then x i new-value 3 if cost > 0 k neighbors received-improve[k] 0 quasi-local opt. 4 then increase weight of constraint violations 5 k neighbors k.handle-ok?(i,x i )

83 Distributed Breakout Example

84 Distributed Breakout Example

85 Distributed Breakout Example

86 Distributed Breakout Example

87 Distributed Breakout Theorem (Distributed Breakout is not Complete) Distributed breakout can get stuck in local optima. Therefore, there are cases where a solution exists and it cannot find it. Proof. By example.

88 Distributed Breakout Theorem (Distributed Breakout is not Complete) Distributed breakout can get stuck in local optima. Therefore, there are cases where a solution exists and it cannot find it. Proof. By example. In practice, its really good.

89 Distributed Constraint Optimization Centralized Outline

90 Distributed Constraint Optimization Centralized Definition (Constraint Optimization Problem (COP)) Given variables x 1,x 2,...x n with domains D 1,D 2,...D n and a set of constraints P of the form pk(x k1,x k2,...,x kj ) R, find assignments for all the variables such that the sum of the constraint values is minimized.

91 Distributed Constraint Optimization Centralized BRANCH-AND-BOUND-COP() 1 c Minimum cost found. Global variable. 2 g Best solution found. Global variable. 3 BRANCH-AND-BOUND-COP-HELPER(1, ) 4 return g BRANCH-AND-BOUND-COP-HELPER(i, g) 1 if i = n 2 then if P(g) < c 3 then g g 4 c P(g) 5 return 6 for v D i 7 do g g + {x i v} 8 if P(g) < c 9 then BRANCH-AND-BOUND-COP-HELPER(i + 1,g )

92 Distributed Constraint Optimization Centralized Definition (Distributed Constraint Optimization Problem (DCOP)) Give each agent one of the variables in a COP. Agents are responsible for finding a value for their variable and can find out the values of their neighbors via communication

93 Distributed Constraint Optimization Adopt Outline

94 Distributed Constraint Optimization Adopt Remote Procedure Calls threshold tell children how much cost they can incur, ignore anything that costs more than that. value tell descendants what value agent sets itself to. cost tell parent lower and upper bounds of cost given the current value assignments of ancestors.

95 Distributed Constraint Optimization Adopt d i d j p(d i,d j ) p x 2 = 1 x 3 = 0 p p p x 1 = 0 x 4 = 1

96 Distributed Constraint Optimization Adopt x 1 = 0 d i d j p(d i,d j ) p p p x 2 = 0 p x 3 = 0 x 4 = 0

97 Distributed Constraint Optimization Adopt x 1 = 0 d i d j p(d i,d j ) value x 1 = 0 x 2 = 0 value x 1 = 0 value x 2 = 0 x 3 = 0 x 4 = 0

98 Distributed Constraint Optimization Adopt d i d j p(d i,d j ) cost 2,2 x 1 = 0 x 2 = 0 x 1 = 0 x 2 = 0 x 3 = 0 x 4 = 0 cost 1, x 1 = 0 cost 1,1 x 2 = 0

99 Distributed Constraint Optimization Adopt x 1 = 1 d i d j p(d i,d j ) value x 1 = 1 x 2 = 0 value x 1 = 1 x 3 = 0 x 4 = 0

100 Distributed Constraint Optimization Adopt d i d j p(d i,d j ) cost 2,2 x 1 = 1 x 2 = 0 x 1 = 1 x 2 = 1 cost 0, x 1 = 1 value x 2 = 1 x 3 = 1 x 4 = 0

101 Distributed Constraint Optimization Adopt d i d j p(d i,d j ) cost 0,0 x 1 = 1 x 2 = 1 x 1 = 1 x 2 = 1 x 3 = 1 x 4 = 1 cost 0,3 x 1 = 1 cost 0,0 x 2 = 1

102 Distributed Constraint Optimization Adopt d i d j p(d i,d j ) x 1 = 1 x 2 = 1 cost 0,0 x 1 = 1 x 3 = 1 x 4 = 1

103 Distributed Constraint Optimization OptAPO Outline

104 Distributed Constraint Optimization OptAPO

105 Distributed Constraint Optimization OptAPO

106 Distributed Constraint Optimization OptAPO

107 Distributed Constraint Optimization OptAPO

108 Distributed Constraint Optimization OptAPO

109 Distributed Constraint Optimization OptAPO

110 Distributed Constraint Optimization OptAPO Theorem (APO worst case is centralized search) In the worst case APO (OptAPO) will make one agent do a completely centralized search of the complete problem space. Proof. By example.

111 Distributed Constraint Optimization OptAPO Adopt versus OptAPO Adopt is better when communications are fast. OptAPO is better when communications are slow. Both have very bad worst-case but seem to perform well.

Termination Problem of the APO Algorithm

Termination Problem of the APO Algorithm Tal Grinshpoun, Moshe Zazon, Maxim Binshtok, and Amnon Meisels Department of Computer Science Ben-Gurion University of the Negev Beer-Sheva, Israel Abstract. Asynchronous