Multiagent Constraint Optimization on the Constraint Composite Graph - PowerPoint PPT Presentation


 
 Multiagent Constraint Optimization on the Constraint Composite Graph Ferdinando Fioretto 
 Hong Xu Sven Koenig T. K. Satish Kumar University of Michigan University of Southern California OptMAS 2018

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Content • Distributed Constraint Optimization Problems • The Constraint Composite Graph (CCG) 
 • CCG-MaxSum • Experimental Evaluation • Conclusions � 3

DCOP | CCG | CCG-MaxSum | Results | Conclusions Distributed Constraint Optimization < X, D, F, A , α >: • X : Set of variables. x 1 x 2 f 1 • D : Set of finite domains for each variable. 0 0 0.5 0 1 0.6 • F : Set of constraints between variables . 1 0 0.7 • A : Set of agents, controlling the variables in X . 1 1 0.3 • α : Mapping of variables to agents. 
 • GOAL: Find a cost minimal assignment. min min F � 4

DCOP | CCG | CCG-MaxSum | Results | Conclusions Distributed Constraint Optimization • Agents coordinate an assignment for their x a variables. • Agents operate distributedly. f ab f ac Communication: x b x c • By exchanging messages. f bc • Restricted to agent’s local neighbors. 
 f bd Knowledge: x d • Restricted to agent’s sub-problem. � 5

DCOP | CCG | CCG-MaxSum | Results | Conclusions DCOP: Algorithms • OPT-APO Search • AFB • ADPOT; BnB-ADPOT Complete • PC-DPOP Inference • DPOP and variants • DSA Search • MGM Incomplete Inference • Max-Sum and variants • D-Gibbs Sampling � 6

DCOP | CCG | CCG-MaxSum | Results | Conclusions DCOP: Representation a 1 x 1 a 1 a 1 x 1 x 1 f 1 f 3 a 2 x 2 a 2 a 2 a 3 a 3 x 2 x 3 x 2 x 3 f 2 a 3 x 3 Constraint Graph Pseudo-Tree Factor Graph � 7

DCOP | CCG | CCG-MaxSum | Results | Conclusions DCOP: Representation a 1 x 1 a 1 a 1 x 1 x 1 f 1 f 3 a 2 This work investigate the use of the CCG, an x 2 a 2 a 2 a 3 a 3 alternative representation, to solve DCOPs x 2 x 3 x 2 x 3 f 2 a 3 x 3 Constraint Graph Pseudo-Tree Factor Graph Assumption: The focus of this talk is restricted to Boolean DCOPs � 8

DCOP | CCG | CCG-MaxSum | Results | Conclusions Constraint Composite Graph Graphical Structure a 1 • DCOP structure: x 1 • Graphical Structure: Interaction of a 2 a 3 cost functions and joint assignments x 2 x 3 • Numerical Structure: Values associated to cost functions 
 Numerical Structure f 1 x 1 x 2 0 0 0.5 • How can we exploit both these 0 1 0.6 structures during problem solving? 1 0 0.7 1 1 0.3 � 9


 
 
 
 
 
 DCOP | CCG | CCG-MaxSum | Results | Conclusions Constraint Composite Graph Graphical Structure • The Constraint Composite Graph (CCG) a 1 [Kumar:08] is a graph 
 x 1 G = ( X ∪ Y ∪ Z, E, w ) a 2 a 3 weights DCOP 
 auxiliary variables variables x 2 x 3 Represents explicitly both structures Numerical Structure • Can be constructed in polytime f 1 x 1 x 2 0 0 0.5 • 
 0 1 0.6 1 0 0.7 1 1 0.3 � 10


 
 
 
 DCOP | CCG | CCG-MaxSum | Results | Conclusions Constraint Composite Graph Graphical Structure • The Constraint Composite Graph (CCG) a 1 [Kumar:08] is a graph 
 x 1 G = ( X ∪ Y ∪ Z, E, w ) a 2 a 3 weights DCOP 
 auxiliary variables variables x 2 x 3 Represents explicitly both structures Numerical Structure • Can be constructed in polytime f 1 x 1 x 2 0 0 0.5 • Solving a DCOP can be reformatted as solving a 0 1 0.6 Minimum Weighted Vertex Cover on its 1 0 0.7 associated CCG 
 1 1 0.3 [extended result from Kumar:16] � 11

DCOP | CCG | CCG-MaxSum | Results | Conclusions The Nemhauser-Trotter (NT) Reduction • Polytime kernelization technique used to a 1 reduce the size of the MWVC x 1 • Minimum Weighted Vertex Cover a 2 a 3 | V | x 2 x 3 X Minimize w i Z i i =1 ∀ v i ∈ V : Z i ∈ { 0 , 1 } ∀ ( v i , v j ) ∈ E : Z i + Z j ≥ 1 � 12

DCOP | CCG | CCG-MaxSum | Results | Conclusions The Nemhauser-Trotter (NT) Reduction • Polytime kernelization technique used to a 1 reduce the size of the MWVC x 1 • Minimum Weighted Vertex Cover a 2 a 3 | V | x 2 x 3 X Minimize w i Z i i =1 ∀ v i ∈ V : Z i ∈ [0 , 1] ⊆ R ∀ ( v i , v j ) ∈ E : Z i + Z j ≥ 1 Z ∈ { 0 , 1 • Relax LP is half integral 2 , 1 } • NT: There is a MWVC that includes v i if Z i =1 and exclude v i if Z i =0 � 13

DCOP | CCG | CCG-MaxSum | Results | Conclusions CCG Construction #1 Construct Polynomial • Each agent, expresses its cost function as a polynomial 1 1 f 1 x 1 x 2 p 1 ( x 1 , x 2 ) = c 00 + c 01 x 1 + c 10 x 2 + c 11 x 1 x 2 . 0 0 0.5 0 1 0.6 Its coefficients can be computed by standard 
 1 0 0.7 Gaussian Elimination so that: 1 1 0.3 p 1 (0 , 0) = 0 . 5 p 1 (0 , 1) = 0 . 6 p 1 (1 , 0) = 0 . 7 p 1 (1 , 1) = 0 . 3 . c 00 = 0.5, c 01 = 0.2, c 10 = 0.1, c 11 = -0.5. � 14

DCOP | CCG | CCG-MaxSum | Results | Conclusions CCG Construction #2 Construct Gadget for each polynomial • Construct a “lifted representation” or a gadget graph for each term of a polynomial of each cost function f 1 1 1 x 1 x 2 p 1 ( x 1 , x 2 ) = c 00 + c 01 x 1 + c 10 x 2 + c 11 x 1 x 2 . 0 0 0.5 0 1 0.6 x 1 x 2 x 1 x 2 1 0 0.7 0.2 0.1 0 0 1 1 0.3 c 10 x 2 c 01 x 1 0.5 y 1 c 00 + c 11 x 1 x 2 � 15

DCOP | CCG | CCG-MaxSum | Results | Conclusions CCG Construction #2 Construct Gadget for each polynomial • Construct a “lifted representation” or a gadget graph for each term of a polynomial of each cost function f 1 1 1 x 1 x 2 p 1 ( x 1 , x 2 ) = c 00 + c 01 x 1 + c 10 x 2 + c 11 x 1 x 2 . 0 0 0.5 0 1 0.6 x 2 x 1 1 0 0.7 0.2 0.1 1 1 0.3 0.5 y 1 c 00 = 0.5, c 01 = 0.2, c 10 = 0.1, c 11 = -0.5. � 16

DCOP | CCG | CCG-MaxSum | Results | Conclusions CCG Construction #2 Construct Gadget for each polynomial • Construct a “lifted representation” or a gadget graph for each term of a polynomial of each cost function f 1 1 1 x 1 x 2 p 1 ( x 1 , x 2 ) = c 00 + c 01 x 1 + c 10 x 2 + c 11 x 1 x 2 . 0 0 0.5 0 1 0.6 x 2 x 1 1 0 0.7 0.2 0.1 1 1 0.3 0.5 y 1 c 00 = 0.5, c 01 = 0.2, c 10 = 0.1, c 11 = -0.5. � 17

DCOP | CCG | CCG-MaxSum | Results | Conclusions CCG Construction #2 Construct Gadget for each polynomial • Construct a “lifted representation” or a gadget graph for each term of a polynomial of each cost function f 1 1 1 x 1 x 2 p 1 ( x 1 , x 2 ) = c 00 + c 01 x 1 + c 10 x 2 + c 11 x 1 x 2 . 0 0 0.5 0 1 0.6 x 2 x 1 1 0 0.7 0.2 0.1 1 1 0.3 0.5 y 1 c 00 = 0.5, c 01 = 0.2, c 10 = 0.1, c 11 = -0.5. � 18

DCOP | CCG | CCG-MaxSum | Results | Conclusions CCG Construction #2 Construct Gadget for each polynomial • Construct a “lifted representation” or a gadget graph for each term of a polynomial of each cost function f 1 1 1 x 1 x 2 p 1 ( x 1 , x 2 ) = c 00 + c 01 x 1 + c 10 x 2 + c 11 x 1 x 2 . 0 0 0.5 0 1 0.6 x 2 x 1 1 0 0.7 0.2 0.1 1 1 0.3 0.5 y 1 c 00 = 0.5, c 01 = 0.2, c 10 = 0.1, c 11 = -0.5. � 19

DCOP | CCG | CCG-MaxSum | Results | Conclusions CCG Construction #2 Construct Gadget for each polynomial • Construct a “lifted representation” or a gadget graph for each term of a polynomial of each cost function f 1 1 1 x 1 x 2 p 1 ( x 1 , x 2 ) = c 00 + c 01 x 1 + c 10 x 2 + c 11 x 1 x 2 . 0 0 0.5 0 1 0.6 x 2 x 1 1 0 0.7 0.2 0.1 1 1 0.3 0.5 y 1 c 00 = 0.5, c 01 = 0.2, c 10 = 0.1, c 11 = -0.5. � 20

DCOP | CCG | CCG-MaxSum | Results | Conclusions CCG Construction #2 Construct Gadget for each polynomial • Construct a “lifted representation” or a gadget graph for each term of a polynomial of each cost function x 1 x i y 1 f 1 x 1 x 2 0 0 0 0 ∞ 0 0 0.5 1 0.2 0 1 0 x 2 1 0 0 0 1 0.6 x 2 x 1 0 0 1 1 0 1 0 0.7 0.2 0.1 1 0.1 i = 1, 2 1 1 0.3 y 1 0 0 1 0.5 0.5 y 1 � 21