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A Distributed, Complete Method for Multi-Agent Constraint Optimization.

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### A Distributed, Complete Method forMulti-Agent Constraint Optimization

Adrian Petcu, Boi Faltingsadrian.petcu@epfl.ch; boi.faltings@epfl.chArtificial Intelligence LaboratorySwiss Federal Institute of Technology (EPFL)IN-Ecublens, 1015 Lausanne, Switzerlandhttp://liawww.epfl.ch/People/apetcu/

• Problem description

• DTREE – tree propagation algorithm

• CyPro – utility propagation with cycles

• CyCopt – splitting the problem

• Conclusions & future work

• Constraint Optimization Problems (COP) – tuple <X, D, R>

• MCOP – multi-agent version of COP: in general, one variable per agent

• MCOP – solutions have a “quality” to be maximized (constraints are utility functions, not predicates)

• Problem description

• DTREE – tree propagation algorithm

• CyPro – utility propagation with cycles

• CyCopt – splitting the problem

• Conclusions & future work

• Distributed utility propagation algorithm

• The leaves of the tree initiate the propagation, and then all nodes forward messages by the “k-1 rule”

• Nodes terminate after receiving all k messages (k neighbors, one message each)

• 2 x (n-1) fixed size messages transmitted.

x0

x4

x1

x3

x6

x7

x5

x2

Done!

X2X1

X1X0

X3X1

R(X1,X0)

• Problem description

• DTREE – tree propagation algorithm

• CyPro – utility propagation with cycles

• CyCopt – splitting the problem

• Conclusions & future work

• Idea: reduce a complex problem (with cycles) to separate cycle-free parts

• Choose a set of cycle cut nodes, and initiate propagation from them.

• Explore all combinations of values of CC nodes, and have them “assemble” the results

“normal” nodes forward by the k-1 rule, and combine contexts

Xj,Xk

Xj,Xk

Xk=vk0

Xj=vj0

Tree parts send messages as before

Cycle cuts act like leaves

• Problem description

• DTREE – tree propagation algorithm

• CyPro – utility propagation with cycles

• CyCopt – splitting the problem

• Conclusions & future work

• Cyclic sub graphs can be processed independently if they are connected through only one CC node.

• A meta-tree is formed by CC nodes and cyclic sub graphs

• Processing in the meta-tree proceeds like in DTREE

Xj

Xk

• Problem description

• DTREE – tree propagation algorithm

• CyPro – utility propagation with cycles

• CyCopt – splitting the problem

• Conclusions & future work

• Linear message size, linear memory

• Reduction in complexity from domn >> dom|CC|>> dom|CCs in largest subgraph|

• Choice of the cycle cutset is important

• Exhaustive search within a cyclic subgraph is maybe not necessary

• F. Kschischang, B.Frey, and H. Loeliger. Factor graphs and the sum-product algorithm. IEEE TRANSACTIONS ON INFORMATION THEORY, 2001.

• Rina Dechter. Constraint Processing. Morgan Kaufmann, 2003.

• P. Modi, W. Shen, M. Tambe, and M. Yokoo. An asynchronous complete method for distributed constraint optimization, 2003.