Distributed Constraint Optimization

1. Distributed Constraint Optimization * some slides courtesy of P. Modi http://www.cs.drexel.edu/~pmodi/

2. Outline • DCOP and real world problems • DiMES • Algorithms to solve DCOP • Synchronous Branch and Bound • ADOPT (distributed search) • DPOP (dynamic programming) • DCPOP • Future work

3. Distributed Constraint Optimization Problem (DCOP) • Definition: • V = A set of variables • Di = A domain of values for Vi • U = A set of utility functions on V • Goal is to optimize global utility • can also model minimal cost problems by using negative utilities • One agent per variable • Each agent knows Uiwhich is the set of all utility functions that involve Vi

4. DiMES • Framework for capturing real-world domains involving joint-activities • {R1,...,RN} is a set of resources • {E1,...,EK} is a set of events • Some ∆ st T*∆ = Tlatest – Tearliest and T is a natural number • Thus we can characterize the time domain as the set Ŧ = {1,...,T} • An event, Ek, is then the tuple (Ak,Lk;Vk) where: • Ak is the subset of resources required by the event • Lk is the number of contiguous time slots for which the resources Ak are needed • Vk denotes the value per time slot of the kth resource in Ak

5. DiMES (cont’) • It was shown in [Maheswaran et al. 2004] that DiMES can be translated into DCOP • Events are mapped to variables • The domain for each event is the time slot at which that event will start • Utility functions are somewhat complex but were able to be restricted to binary functions • It was also shown that several resource allocation problems can be represented in DiMES (including distributed sensor networks)

6. Synchronous Branch and Bound • Agents are prioritized into a chain (Hirayama97) or tree • Root chooses value, sends to children • Children choose value, evaluate partial solution, send partial solution (with cost) to children • When cost exceeds upper bound, backtrack • Agent explores all its values before reporting to parent

7. SyncBB Example

8. Pseudotrees • Solid line • parent/child relationship • Dashed line • pseudo-parent/pseudo-child relationship • Common structure used in search procedures to allow parallel processing of independent branches • A node can only have constraints with nodes in the path to root or with descendants

9. ADOPT • SyncBB backtracks only when suboptimality is proven (current solution is greater than an upper bound) • ADOPT’s backtrack condition – when lower bound gets too high • backtrack before sub-optimality is proven • solutions need revisiting • Agents are ordered in a Pseudotree • Agents concurrently choose values • VALUE messages sent down • COST messages sent up only to parent • THRESHOLD messages sent down only to child

10. ADOPT Example • Suppose parent has two values, “white” and “black”

11. DPOP • Three phase algorithm: • Pseudotree generation • Utility message propagation bottom-up • Optimal value assignments top-down

12. DPOP: Phase 1

13. DPOP: Phase 2 • Propagation starts at leaves, goes up to root • Each agent waits for UTIL messages from children • does a JOIN • sends UTIL message to parent • How many total messages in this phase?

14. DPOP: Phase 2 (cont’) • UTIL Message • maximum utility for all value combinations of parent/pseudo-parents • includes maximum utility values for all children

15. DPOP: Phase 3 • Value Propagation • After Phase 2, root has a summary view of the global UTIL information • Root can then pick the value for itself that gives the best global utility • This value is sent to all children • Children can now choose their own value, given the value of the root, that optimizes the global utility • This process continues until all nodes are assigned a value

16. DCOP Algorithm Summary • Adopt • distributed search • linear size messages • worst case exponential number of messages • with respect to the depth of the pseudotree • DPOP • dynamic programming • worst case exponential size messages • with respect to the induced width of the pseudotree • linear number of messages

17. Can we do better? • Are pseudotrees the most efficient translation? • The minimum induced width pseudotree is currently the most efficient known translation • Finding it is NP-Hard and may require global information • Heuristics are used to produce the pseudotrees • Current distributed heuristics are all based on some form of DFS or BestFS • We prove in a recent paper that pseudotrees produced by these heuristics are suboptimal

18. Cross-Edged Pseudotrees • Pseudotrees that include edges between nodes in separate branches • The dashed line is a cross-edge • This relaxed form of a pseudotree can produce shorter trees, as well as less overlap between constraints

19. DCPOP • Our extension to DPOP that correctly handles Cross-Edged Pseudotrees • We have proved that using an edge-traversal heuristic (DFS, BestFS) it is impossible to produce a traditional pseudotree that outperforms a well chosen cross-edged pseudotree • Edge-traversal heuristics are popular because they are easily done in a distributed fashion and require no sharing of global information

20. DCPOP (cont’) • Computation size is closer to the minimum induced width than with DPOP • Message size can actually be smaller than the minimum induced width • A new measurement of sequential path cost (represents the maximal amount of parallelism achieved) also shows improvement

21. DCPOP Metrics

22. Future Work • DCOP mapping for a TAEMS based task/resource allocation problem • Full integration of uncertainty characteristics into the DCOP model • Anytime adaptation with uncertainty