1. Allocating a Timeshared Resource: An Application to Grid Computing Alan M. Frisch Artificial Intelligence Group Dept. Computer Science University of York In collaboration with: Mark Bartlett (York), Youssef Hamadi (MSR), Ian Miguel(St.Andrews), Armagan Tarim (Cork) Chris Unsworth (Glasgow) With funding from: Microsoft Research

2. Introduction Effective use of grid computing will require efficient methods to solve a variety of combinatorial problems, such as: Resource allocation, Configuration, and Scheduling. Grid resources are not currently reserved in advance, but will in future.

3. Advanced Reservation This talk considers to the problem of allocating a time-shared or space-shared resource to bids for advanced reservations. –Communication bandwidth –Computer memory or disk space –Equivalent rooms in a hotel that handles block booking Use combinatorial optimisation methods –Constraint programming –Operations research: optimisation –Artificial intelligence: search

4. Simple Model of Advanced Reservation Given bids for quantities of the resource, each offering a price: Accept a subset of the bids that can be fulfilled and maximises revenue. This talk presents an algorithm for solving large instances of this problem in batch mode.

5. Given: –A finite set of times totally ordered by  –Along with each time t, capacity(t), a positive integer –A finite set of bids –Along with each bid b price(b), demand(b), two positive integers duration(b), a non-empty interval of times Find: a set accept  bids Such that:  t  times  demand(b)  capacity(t) Temporal Knapsack Problem (TKP) b  accept|t  duration(b) Maximising:  price(b) b  accept

6. Sample Instance of TKP Bid3: 5 units, £20 t2t3t4t5t1 Uniform Capacity: 10 Bid2: 6 units, £10 Bid1: 6 units, £11 t2t3t4t5t1 Bid2: 6 units, £10 Bid1: 6 units, £11 Solution: Objective Fn = £21

7. TKP Generalises Knapsack Every instance of Knapsack is an instance of TKP in which |times| = 1. Hence TKP is NP-hard. (Easy to see that TKP is NP-easy)

8. Multi-dimensional Knapsack (MDK) generalises TKP Multi-dimensional knapsack –each item has n-dimensional vector as its size –knapsack has n-dimensional vector as its capacity –sum of sizes of items in knapsack cannot exceed capacity of knapsack Every instance of TKP is an instance of MDK in which the times are the dimensions and every item has size of form (0 … 0 demand … demand 0 … 0)

9. TKP Reduces to Integer Linear Programming Bid3: 5 units, £20 t2t3t4t5t1 Uniform Capacity: 10 Bid2: 6 units, £10 Bid1: 6 units, £11 Maximise:   X b  price(b) b  bids Subject to:  b X b in {0,1} 6X 2  6X 2 + 5X 3  5X 3   6X 1 + 5X 3  6X 1 

10. Solving the TKP Decomposition algorithm Greedy Algorithm (approximation) Integer Linear Programming: CPLEX

11. Decomposition Algorithm

12. Makes use of three operations: Reduce the problem into a simpler one Split into independent sub-problems. Branch on accept/reject a bid. These operations are composed into a branch and bound search for optimal solutions.

13. Reduce: Forced Reject If at any time during the interval required by a bid, available bandwidth is less than that requested, bid rejected. 10 units required. Bid: Available Capacity: 10 5 t1t2t3t4t5t6t7t8t9

14. Reduce: Remove Times Bid3: 5 units t5t6 t7t8t9t10t11t12t1t2t3t4 Capacity: 10 Demand: Bid2: 6 units Bid1: 6 units t5t8 5 6 6 Problem reduced to important decision points.

15. Reduce: Remove Times  Forced Accept Bid3: 5 units t5t6 t7t8t9t10t11t12t1t2t3t4 Capacity: 10 Demand: Bid2: 6 units Bid1: 4 units t4t5 5 6 Bid1 Accepted.

16. Split Can solve independently, combine solutions. Bid3: 5 units t5t6 t7t8t9t10t11t12t1t2t3t4 Bid2: 6 units Bid1: 6 units t5t1t2t3t4 Bid2: 6 units Bid3: 5 units t6 t7t8t9t10t11t12 Bid1: 6 units AND

17. Branch t5t8 5 6 6 t5t8 6 6 Capacity 10 Demand 11 t5t8 6 6 Reject Blue Accept Blue OR Demand 6 Capacity 5 Capacity 10 Demand 6

18. Interaction: Example Bid 1: Forced reject. Bid2: 5 units t5t6 t7t8t9t1t2t3t4 Bid1: 6 units Capacity: Demand:

19. Interaction: Example Times removed. So bid 2 accepted. Bid2: 5 units t5t6 t7t8t9t1t2t3t4 Capacity: Demand:

20. Decomposition Algorithm: Initialisation(P) Reduce(P) Split P into set of problems S. Foreach s  S, Solve(s).

21. Decomposition Algorithm: Solve(P) If no bids, return. Select a bid b from bids. Branch on: 1.Reject(b) then Reduce(P). Split(P) into set of problems S. Foreach s  S, Solve(s). 2.Accept(b) then Reduce(P). Split(P) into set of problems S. Foreach s  S, Solve(s).

22. Deterministic Algorithm: AND/OR Search Space A,B,C,D,E,F B,E,FC,D,E,F E,F C,D B,E,F AND Split Reject A  Accept B Accept A  Reject C,D Split

23. A Solution is a Tree A,B,C,D,E,F B,E,F E,F C,D B,E,F AND Split Reject A  Accept B Accept A  Reject C,D Split

24. Growing the AND/OR Tree While tree contains an OR node with bids do –Choose one such OR node, N –Choose a bid in N; call it b –Generate two children of N by branching on b –Reduce each child –Split each child How to choose node to expand, N –Or partially-grown solution tree How to choose bid to branch on, b

25. Propagating Bounds A,B,C,D,E,F B,E,FC,D,E,F E,F C,D B,E,F AND Split Reject A  Accept B (11) Accept A (10)  Reject C,D     Split 

26. Using Bounds: Branch and Bound A,B,C,D,E,F B,E,FC,D,E,F E,F C,D B,E,F AND Split Reject A  Accept B (11) Accept A (10)  Reject C,D     Split 

27. Using Bounds: Branch and Bound A,B,C,D,E,F B,E,FC,D,E,F E,F C,D B,E,F AND Split Reject A  Accept B (11) Accept A (10)  Reject C,D     Split 

28. Choosing a Node: Search Strategy AO* –Expand tree with highest upper bound –Admissible: first solution found is best –Optimal: expands fewest possible nodes for any given upper-bound method –Uses a lot of memory Depth First –Expand deepest node –Uses little memory, but expands more nodes than AO*

29. Choosing a Bid to Branch on Longest Duration: choose bid with longest duration –Intuition: maximise reduction; get the most information from one choice Force splits: choose adjacent times and branch on all bids that span those two times. –Consequence: we can split problem between the two times. –Intuition: maximise splitting –Trade-off between splitting into equal subproblems and minimising number of branches. In our experiments forced splits always outperformed longest duration.

30. Performance

31. Compare Performance of Three Solvers Decomposition algorithm with depth-first search. Decomposition algorithm with AO* search. CPLEX on ILP formulation

32. Generating Random Instances Given parameters: max-rate, number of bids Allocate resources for a period of one month. Granularity of 15 minutes, gives 2,880 times. Start time of each bid: uniform random [1, 2880] Duration of each bid: uniform random [1, 100] Bandwidth of each bid: uniform random [1, 50] Rate of each bid: uniform random [1, max-rate] Price and end time of each bid derived from above.

33. Experimental Results Number of bids: 400, 450, 500, 550, 600 MaxRate: 1.0, 1.2, 1.4, 1.6, 1.8, 2.0 As MaxRate increases, the instances become harder for all solvers.

34. Experimental Results Now consider performance as num of bids varies. Use force-splits for decomposition algorithm For each number of bids, used a suite of instances with 20 members from each max-rate value. Up to 550 orders: all algorithms less than 10 secs. Over 550, AO* runs out of memory. Over 600, depth-first decomposition too slow. Over 650, CPLEX runs out of memory.

35. Summary and Future Directions

36. Summary Defined TKP and identified it as a model of advanced reservation Presented special-purpose algorithm for TKP –Decomposition is distinctive feature

37. Future Directions: Improve Current Solver Employ a search algorithm that takes middle ground between time-hungry, space-efficient DFS time-efficient, space-hungry AO* – adapt a space-bounded search algorithm to AND/OR Better heuristic for choosing where to force splits Careful tradeoff between advantages of –minimising branching needed to force a split –splitting into equal-sized subproblems Better data structures –Identify more redundant times

38. Future Directions: Improve Model of Advanced Reservation Allow flexibility in time requested by a bid Accommodate multiple resources and combined bids Dynamic environment –Bids arrive as jobs are being executed –Some jobs can be thrown out or rescheduled, perhaps with a penalty –Some jobs may terminate early

39. Further Information www.cs.york.ac.uk/aig/constraints/Grid Frisch@cs.york.ac.uk