Asynchronous Forward-Bounding for DisCOPs

1. Asynchronous Forward-Bounding for DisCOPs Amir Gershman, Amnon Meiselsand Roie Zivan Dept. of Computer Science Ben-Gurion University ECAI-06 - Asynchronous Forward-Bounding

2. Constraint Optimization Problems • variables, each with a finite domain of values • Variables are connected by constraints • Constraints assign a non-negative cost to every value assignment of the constrained variables – Weighted CSPs • A solution is a full assignment (to all variables) with minimal accumulated cost a g 1 b h 2 c k 5 2 3 d e f ECAI-06 - Asynchronous Forward-Bounding

3. Distributed Problem – Variables held by agents. Each agent knows only the constraints involving its variables Subclasses – MaxCSP, DisMaxCSP Distributed Constraint Optimization Problems a g 1 b h 2 c k 5 2 3 d e f ECAI-06 - Asynchronous Forward-Bounding

4. Branch and Bound X1=a Upper-Bound =∞ Upper-Bound =1 Upper-Bound =2 Lower-Bound = 2 Lower-Bound = 1 Lower-Bound = 0 X2=f X2=e X2=d X3=v2 X3=v1 X3=v2 X3=v1 X4=w2 X4=w1 X4=w1 X4=w2 X4=w1 X4=w2 ECAI-06 - Asynchronous Forward-Bounding

5. Evidence from Centralized B & B • Centralized B&B can be improved by maintaining local consistency [Larrosa Meseguer and Schiex (96,99,2004)] • Several methods (DAC,NC*,AC*) show a dramatic performance improvement over standard B&B • Consistency maintenance produces a “phase-transition” for extremely tight problems (high p2) • Only for a strong enough consistency method [Larrosa, Schiex 04] ECAI-06 - Asynchronous Forward-Bounding

6. ADOPT • Asynchronous complete algorithm for DisWCSP • Uses a pseudo-tree • Agents send their value assignment down the tree and report lower and upper bounds up the tree • ADOPT is “greedy” - values may be revisited • Thresholds were added to prevent thrashing – requires a heuristic • Observed in our empirical evaluation – number of messages increases exponentially with time • Not easy to fix - asynchronous updates ECAI-06 - Asynchronous Forward-Bounding

7. Asynchronous Forward Bounding (AFB) – Main ideas • Synchronous B&B – sequential assignments by agents • Asynchronous maintain distributed local consistency – bounds computed concurrently with assignments ECAI-06 - Asynchronous Forward-Bounding

8. AFB – the framework • A CPA message carries the Current Partial Assignment • Agents extend the CPA by adding an assignment to their variable and passing the CPA forward to the next agent • When an agent exhausts its value domain, it backtracks by undoing its assignment and sending the CPA to the previous agent ECAI-06 - Asynchronous Forward-Bounding

9. A1 A3 2 a g 1 b h c k A2 5 2 3 d CPA = {A1 = a}CPA.cost = 0 e f ECAI-06 - Asynchronous Forward-Bounding

10. A1 A3 2 a g 1 b h c k A2 5 2 3 d e f CPA = {A1 = a, A2=d}CPA.cost = 0 ECAI-06 - Asynchronous Forward-Bounding

11. A1 A3 2 a g 1 b h c k A2 5 2 3 d e CPA = {A1 = a, A2=d, A3=g}CPA.cost = 5 f ECAI-06 - Asynchronous Forward-Bounding

12. AFB – Asynchronous forward-bounding • After assigning (extending) the CPA, copies, FB-CPA, are sent forward to all unassigned agents • Agents receiving a FB_CPA, compute a Lower Bound on the cost of assigning a value to their variable and report it back to the sending agent ECAI-06 - Asynchronous Forward-Bounding

13. A1 A7 A3 A8 A6 a 2 v1 g v1 v1 … 1 b v2 h v2 v2 c v3 k v3 v3 5 A2 2 3 d e CPA = {A1 = a, A2=d}CPA.cost = 0 f FB_CPA = {A1 = a, A2=d}cost = 0 ECAI-06 - Asynchronous Forward-Bounding

14. A7 A6 A6 - Accumulated cost v1 v2 2 5 v1 4 v3 7 v2 9 3 v3 4 2 1 6 A8 v1 FB_CPA = {A1 = a, A2=d}cost = 0 v2 v3 ECAI-06 - Asynchronous Forward-Bounding

15. A7 A6 local cost + forward look v1 v2 2 5 + 1 v1 4 v3 7 + 0 v2 9 3 + 2 v3 4 2 LB6 = min{6,7,5} = 5 1 6 A8 FB_CPA = {A1 = a, A2=d}cost = 0 v1 v2 FB_EST = {A1 = a, A2=d}LB6 = 5 v3 ECAI-06 - Asynchronous Forward-Bounding

16. AFB - backtracking • Early detection of the need for backtrack: • Cost of CPA + Lower Bounds on future assignments ≥ Upper Bound • If this condition holds, the agent changes its assignment to the next possible value, or backtracks if its domain of values is exhausted ECAI-06 - Asynchronous Forward-Bounding

17. CPA = {A1 = a, A2=d}CPA.cost = 0 A2 FB_EST = {A1 = a, A2=d}LB3 = 1 d e FB_EST = {A1 = a, A2=d}LB4 = 2 f FB_EST = {A1 = a, A2=d}LB5 = 1 0 + {1+2+1+5} ≥ UB ? Yes -> create No -> do nothing CPA = {A1 = a, A2=e}CPA.cost = 0 FB_EST = {A1 = a, A2=d}LB6 = 5 ECAI-06 - Asynchronous Forward-Bounding

18. AFB – Determining Relevance • Several CPAs can “co-exist” in the system at the same time (as a result of backtracking) • Need to know which CPAs are obsolete and discard them • Each message includes a timestamp • Receiving agents can determine relevancy • Timestamp mechanism of [Nguyen et al. 2004] • Used also for dynamically ordered ABT [Zivan et al. 2005,6] ECAI-06 - Asynchronous Forward-Bounding

19. AFB – Summary • A CPA travels between agents, assignments are added sequentially • Concurrently, bounds are gathered to ensure the CPA’s consistency • If inconsistent – new CPA generated, with higher timestamp, and search continues • Agents receiving a higher timestamp eventually “kill” (received) obsolete CPAs • An agent reaching a full assignment – globally broadcasts the new upper bound ECAI-06 - Asynchronous Forward-Bounding

20. Experimental Evaluation • Random distributed Max-CSPs • Size: • 10 Variables • 10 Values • Density: p1 = 0.4 • Tightness: p2 = 0.4 – 0.99 • Measures: • Total number of messages • Number of non-concurrent computation steps • In ADOPT - a step includes reading and processing ALL the messages in the agent’s buffer, in order to increase performance ECAI-06 - Asynchronous Forward-Bounding

21. Comparing run-time ECAI-06 - Asynchronous Forward-Bounding

22. Zoom in ECAI-06 - Asynchronous Forward-Bounding

23. Communication load ECAI-06 - Asynchronous Forward-Bounding

24. Concurrency of Computation • AFB – Forward-Bounds by unassigned agents are computed while the CPA accumulates a larger partial assignment • ADOPT – arises from pseudo-tree ordering • good ordering = wide, low depth tree -> increases concurrency and performance • In dense (high p1) problems parallelism suffers • In the extreme, p1=1.0, the tree is a chain ECAI-06 - Asynchronous Forward-Bounding

25. Conclusions • Previous studies measured scalability – increasing number of agents. Present study measures performance by increasing problem difficulty (tightness) • ADOPT’s performance, on MAX-DCOP, increases exponentially while AFB shows a “phase-transition” • AFB’s performance is a result of maintaining local consistency. Similar to other algorithms in the centralized case ECAI-06 - Asynchronous Forward-Bounding

26. Conclusions (II) • Unreported exponential growth in ADOPT’s communication load was observed • AFB is simpler than ADOPT Easier to extend and implement ECAI-06 - Asynchronous Forward-Bounding

27. Future Work • Extending AFB – • Value / Variable orderings • Stronger form of consistency • Empirical evaluation of WCSPs • Full spectrum of density (p1) and tightness (p2) • How to select costs ? ECAI-06 - Asynchronous Forward-Bounding

28. Thank you ECAI-06 - Asynchronous Forward-Bounding