1. CS584 - Software Multiagent Systems Lecture 12 Distributed constraint optimization II: Incomplete algorithms and recent theoretical results

2. University of Southern California 2 Personal Assistant Agents for Scheduling Sensor Networks Multi- Spacecraft Coordination Distributed Constraint Optimization DCOP (Distributed Constraint Optimization Problem) –Agents cooperate to assign values to variables, subject to constraints, to maximize a global objective. –R(a)=∑ R S (a) for all constraints S

3. University of Southern California 3 Algorithms for DCOP Complete algorithms –Pro: finds optimal solution –Con: not always feasible (exponential in time or space) Local algorithms (today) –Pro: usually finds a high-quality solution quickly –Con: not optimal (but can guarantee within % of optimal)

4. Part I Algorithms

5. University of Southern California 5 k-optimality k-optimal solution (k-optimum) –No deviation by ≤ k agents can increase solution quality –Local optimum –Globally optimal solution: 000. –1-optimal solutions: 000, 111 –k-optimal algorithms: DSA (k=1), MGM-2 (k=2),...

6. University of Southern California 6 Approach Decompose DCOP. Each agent only sees its local constraints. Agents maximize individual utility. –(individual view of team utility)

7. University of Southern California 7 1-optimal algorithms Algorithms for 1 agent: –Broadcast current value to all neighbors and receive neighbors’ values. –Find new value that gives highest gain in utility, assuming that neighbors stay fixed. –Decide whether or not to change value, and act accordingly.

8. Algorithms 2 Variations –MGM (Maximum Gain Message) Modified DBA (Yokoo) Exchange messages with neighbors about utility gains. Neighbor that can make highest gain gets to move. –DSA (Distributed Stochastic Algorithm) Fitzpatrick and Meertens Move according to probability p. University of Southern California 8

9. 9 x1x2x3 +50 0+8 8108 x1x2x3 000 MGM – Maximal Gain Message Monotonic algo., but gets stuck in a local optimum! Only one agent in a neighborhood moves at a time

10. University of Southern California 10 x1x2x3 +50 0+8 8108 x1x2x3 5165 DSA – Distributed Stochastic Algorithm One possible path (say p=0.5):.1.2.3 x1x2x3 000.6.1.7

11. University of Southern California 11 Experimental Domains Regular graph coloring (~sensor network) –Cost if neighbors choose same value. Randomized DCOP –Each combination of neighbors’ values gets uniform random reward High-stakes scenario (~UAVs) –Large cost if neighbors choose same value. –Otherwise, small uniform random reward is given. –Add “safe” value where all agents start. No reward or penalty if neighbors choose this value.

12. University of Southern California 12 DSA vs. MGM Graph coloring and Randomized DCOP: –DSA gives higher solution quality than MGM. –DSA improves more quickly than MGM. High-stakes scenario: –DSA and MGM give same solution quality. –MGM generally improves more quickly than DSA. But, these graphs are averages....

13. University of Southern California 13 DSA vs. MGM MGM increases monotonically Much better for –anytime algorithm –high-stakes domains.

14. University of Southern California 14 Algorithms with higher k Until now (DSA, MGM), agents have only acted based on their own, local constraints –a myopic worldview Now, we look at algorithms where agents form groups, and act based on all constraints in the group. –enlarging the worldview First step: groups of 2. –“2-optimality” –Maheswaran, Pearce, and Tambe ‘04

15. University of Southern California 15 Coordinated Algorithms All agents are either offerers or receivers with probability q. Offerers: –Pick neighbor j at random, and calculate my gains from all combinations of values from myself and j. –Send this information (several offers) as a message to j.,…> Receivers: –Accept the offer that makes my group’s gain the highest, or just move alone instead. –groupGain = offerersGain + receiversGain - gain in common link. –If I accept an offer, tell the offerer which one I am accepting, and how much our group will gain.

16. University of Southern California 16 2-optimal algorithms To improve solution quality, agents can form groups of 2 Groups move according to group utility –sum of all constraints on any group member 2-optimal algorithm –any connected group of 2 agents can coordinate to make a joint move. 2-optimum –state at which no group of up to 2 agents can make a joint move that will increase group reward. Any 2-optimum is also a 1-optimum

17. University of Southern California 17 MGM-2 Form groups of 2 agents and then do: –Send my gain (can be group gain) to all my neighbors. –Receive gain messages from my neighbors. –If I am involved in an accepted offer, If my gain > neighbors’ gain (not counting my partner), send “yes” to my partner. If not, then send “no” to my partner. If I sent “yes” and got “yes”, then make the move in the offer. –If I am not involved in an offer If my gain > neighbors’ gain, then make my best move. 5 message cycles per move –(offer, accept, gain, confirm, move). Monotonically increasing solution quality

18. University of Southern California 18 x1x2x3 +50 0+12 x1x2x3 MGM-2 Example x1x2x3 no gains offerer receiver offerer x1, x2 gain=7 x2, x3 gain=7 x1x2x3 accepts x1, x2 group gain=2 x1x2x3 receiver offerer receiver x2, x3 gain=12 accepts x2, x3 group gain=12

19. University of Southern California 19 SCA-2 (Stochastic Coordination Algorithm) Based on DSA If offerer –Send out offers to a randomly chosen neighbor. –If offer accepted, prepare to do the move in the offer. –If offer not accepted, prepare to move alone (pick move with highest individual gain). If receiver –If accepting offer, send acceptance message back to offerer, and prepare to do the move in the offer. –Else, prepare to move alone. Move, according to probability p. 3 message cycles per move (offer, accept, move).

20. University of Southern California 20 Experimental Trends Monotonic (1-opt, 2-opt) Stochastic (1-opt, 2-opt)

21. Part II Theoretical Results

22. University of Southern California 22 Guarantees on Solution Quality Guarantee of k-optimum as % of global optimum –Factors: k (how local of an optimum) m (maximum -arity of constraints) n (number of agents) constraint graph structure (if known) Note: actual costs/rewards on constraints –distributed among agents, not known a priori

23. University of Southern California 23 Guarantees on Solution Quality Three results Guarantees for: –Fully-connected DCOP graphs Applies to all graphs (i.e. when graph is unknown) Closed-form equation –Particular graphs Stars Rings Closed-form equation –Arbitrary DCOP graphs Linear program

24. University of Southern California 24 Fully-Connected Graph Reward of k-optimum in terms of global optimum Independent of rewards Independent of domain size Provably tight (in paper) One assumption: rewards are non-negative For binary graph (m=2),

25. University of Southern California 25 Proof sketch / example 10R(a) ≥ ∑R(â) 10R(a) ≥ 3R(a*) + 1R(a) Fully connected graph n = 5 agents m = 2 (binary constraints) k = 3 Goal: express R(a) in terms of R(a*). a dominates: Â = {11100 11010 11001 10110 10101 10011 01110 01101 01011 00111} a* = 11111 (global opt) a = 00000 (3-opt)

26. University of Southern California 26 Other graph types Ring: Star: Similar analysis, but exploit graph structure –Only consider  where connected subsets of k agents deviate

27. University of Southern California 27 Proof sketch / example 5R(a) ≥ ∑R(â) 5R(a) ≥ 2R(a*) + 1R(a) Ring graph n = 5 agents m = 2 (binary constraints) k = 3 Goal: express R(a) in terms of R(a*). a dominates: Â = {11100 01110 00111 10011 11001} a* = 11111 (global opt) a = 00000 (3-opt)

28. University of Southern California 28 Arbitrary graph Arbitrary graph = linear program Minimize R(a)/R(a*) such that: –for all dominated assignments â, R(a) - R(â) ≥ 0. Each constraint S in DCOP = 2 variables in LP. –1: R S (a) for reward on S in k-optimal solution –2: R S (a*) for reward on S in global optimum –All other rewards on S taken as 0 (as before). Why ok? R(a) and R(a*) don’t change a still k-optimal (no k agents would change) a* still globally optimal

29. University of Southern California 29 Arbitrary graph Minimize R(a)/R(a*) such that: –for all dominated assignments â, R(a) - R(â) ≥ 0. R(a) = ∑ R S (a) over all constraints S. R(a*) = ∑ R S (a*) over all S. Every R(â) = ∑ R S (a) + ∑ R S (a*) for various S. Why? –Consider assignment 00011 in 5-agent ring. R(00011) = R {1,2} (00011) + R {2,3} (00011) + R {3,4} (00011) + R {4,5} (00011) + R {5,1} (00011) R {1,2} (a) + R {2,3} (a) + 0 + R {4,5} (a*) + 0 Minimize ∑ R S (a) / ∑ R S (a*) –Subject to linear inequalities on R S (a) and R S (a*) for various S. Linear Fractional Program: reducible to LP (textbook)

30. University of Southern California 30 Experimental Results Designer can choose appropriate k or topology!

31. University of Southern California 31 Experimental Results

32. University of Southern California 32 Conclusions Guarantees for k-optima in DCOPs as % of optimum –Despite not knowing constraint rewards –Helps choose algorithm to use –Helps choose topology to use Big idea: –Single agent: Rationality -> Bounded Rationality –Multi agent: Global Optimality -> k-Optimality –Ability to centralize information (coordinate) is bounded (only groups of k agents) –Guarantees on performance of “bounded coordination”

33. University of Southern California 33 Readings J. P. Pearce and M. Tambe, "Quality Guarantees on k- Optimal Solutions for Distributed Constraint Optimization Problems," in IJCAI-07. R. T. Maheswaran, J. P. Pearce and M. Tambe, "Distributed Algorithms for DCOP: A Graphical-Game-Based Approach," in PDCS-04. –(just read algorithms - don’t need to read proofs) W. Zhang, Z. Xing, G. Wang and L. Wittenburg, "An analysis and application of distributed constraint satisfaction and optimization algorithms in sensor networks," in AAMAS- 03.