Tung-Wei Kuo , Kate Ching-Ju Lin, and Ming- Jer Tsai Academia Sinica , Taiwan

1. Maximizing Submodular Set Function with Connectivity Constraint: Theory and Application to Networks Tung-Wei Kuo, Kate Ching-JuLin, and Ming-JerTsai Academia Sinica, Taiwan National TsingHua University, Taiwan

2. Motivation • Mesh network deployment

3. Motivation • Mesh network deployment Candidate location How should we deploy the network?

4. Motivation • Mesh network deployment Candidate location The budget is limited!

5. Connectivity • Only one router can access the Internet • Mesh networks exploit multi-hop relays Candidate location

6. Connectivity • Only one router can access the Internet • Mesh networks exploit multi-hop relays Candidate location

7. Connectivity • Only one router can access the Internet • Mesh networks exploit multi-hop relays The network must be connected!

8. VariousPerformance Metrics • A variety of performance metrics • The number of covered users, total throughput, the size of the coverage area, … Given limited resources (routers or budget), deploy a connected mesh that optimizesthe performance metric

9. Mesh Deployment Problem • Given: • routers, where one of them is a gateway • The set of candidate locations, • The set of connection edges, • The optimization goal (e.g., the number of covered users) A graph This is the optimal solution GOAL: Construct a connected network such that the optimization goal is achieved

10. Our goal: A universal algorithmfor a family of problems whose objective can be modeled as asubmodular set function Design an algorithm for each of the various optimization goals? Many optimization goals can be modeled assubmodular set functions

11. Submodular Set Function A function is a submodular set function if

12. Example: Number of covered users

13. Example: Number of covered users

14. Example: Number of covered users

15. Example: Number of covered users

16. Example: Number of covered users

17. Example: Total Data Rate

18. Example: Total Data Rate

19. Formal Problem Definition • Given: • A graph • A positive integer • A nondecreasing submodular set function onthe set of subsets of with • Goal: Find a subsetsuch that • Connectivity:is connected with respect to • Limited resources: • Optimization goal:is maximized

20. Formal Problem Definition • Given: • A graph • A positive integer • A nondecreasing submodular set function onthe set of subsets of with • Goal: Find a subsetsuch that • Connectivity:is connected with respect to • Limited resources: • Optimization goal:is maximized The problem is NP-hard.An approximation algorithm will be given

21. Our Algorithm

22. The Idea For every candidate location, , generate a solution in the following way: Step 1. Find an area, , centered at Step 2. Deploy some routers on Step 3. Use the remaining routers tomake the solution connected The best solution is then the final output

23. The Solution-Step 1 1. Find an area centered at with a radius of hops

24. The Solution-Step 1 1. Find an area centered at with a radius of hops • radius : hops

25. The Solution-Step 2 2. Deploy routers, where one of them is at the center • radius : hops

26. The Solution-Step 2 # of covered users 2. Deploy routers, where one of them is at the center • radius : hops

27. The Solution-Step 2 # of covered users 2. Deploy routers, where one of them is at the center • radius : hops Candidate location User

28. The Solution-Step 2 # of covered users 2. Deploy routers, where one of them is at the center • radius : hops Candidate location User

29. The Solution-Step 3 # of covered users 3. Use shortest pathsto connect routers to the center • radius : hops Candidate location User

30. The Solution-Step 3 # of covered users 3. Use shortest pathsto connect routers to the center • radius : hops Candidate location User This is a feasible solution

31. The Algorithm For every candidate location, , generate a solution in the following way: Step 1. Find an area, , centered at , with radius Step 2. Deploy routers on Step 3. Use the remaining routers tomake the solution connected How, exactly, should we deploy the routers? The best solution is then the final output.

32. How to Deploy the Routers? • Solvea subproblem that is similar to the main problem, except that • The solution can be disconnected • The center of the given area must be chosen • It is still NP-hard • When is dropped, Nemhauseret al. propose an -approximation algorithm [9] • We modify Nemhauser’salgorithm to satisfy [9] G. L. Nemhauser, L. A. Wolsey, and M. L. Fisher, “An analysis of approximations for maximizing submodular set functions-I,” Mathematical Programming, vol. 14, pp. 265–294, 1978.

33. Approximation Ratio • is the optimal solution when only routers can be used Our algorithm is an -approximation algorithm

34. The Problem with HeterogeneousDeployment Costs Different locations might have different deployment costs

35. Formal Problem Definition • Given: • A vertex-weighted graph • A nondecreasing submodular set function on the set of subsets of with • A positive integer • Find a subset such that: • Connectivity: is connected with respect to • Limited budget:The total weight of • Optimization goal: is maximized

36. Approximation Ratio • , where is the maximum degree of • A special case: Unit disk graph ⇒

37. Simulation Results-Use Synthesis Data

38. Simulation Setting • Field size: 1200 m × 1200 m • User: • # of users: 200 • Zipf’s law • 802.11b • Candidate locations: • Grid network • Grid size: 100 m × 100 m • Communication range: 150 m • Channel error model: 802.11b PHY Simulink Model

39. Another Common Scenario • In some applications, a specific location may need to be included in the solution • We modify our algorithm accordingly: How to find the center? Try all the possible centers and • Ouralgorithm choose the best one Let the specific location be the • Our algorithmw/ specific center desired center

40. Comparison Schemes • Two greedy heuristics: • Try all the possible starting locations • Add one neighboring vertex at a time • Minimum deployment cost or maximum performance gain • When Goal = maximum number of covered users Homogeneous costs We compare with Vandin’s algorithm [17] [17] F. Vandin, E. Upfal, and B. J. Raphael, “Algorithms for detecting significantly mutated pathways in cancer,” Journal of Computational Biology, vol. 18, pp. 507–522, 2011.

41. Simulation Scenarios • Two types of deployment costs: • Homogeneous costs • Heterogeneous costs • Two performance metrics: • Total data rate • The number of covered users

42. Maximum Total Data Rate • Homogeneous costs Total data rate of covered users (Mb/sec) Upper bound Arbitrary solution Greedy: max date rate Greedy: max data rate w/ specific center Our algorithm Our algorithm w/ specific center Number of routers, k

43. Maximum Total Data Rate • Heterogeneous costs Total data rate of covered users (Mb/sec) Upper bound Arbitrary solution Greedy: min cost Greedy: min cost w/ specific center Greedy: max data rate Greedy: max data rate w/ specific center Our algorithm Our algorithm w/ specific center Total budget for deployment, B

44. Maximum Number of Covered Users • Homogeneous costs Upper bound Arbitrary solution Vandin’s algorithm Vandin’salgorithm w/ specific center Our algorithm Our algorithm w/ specific center Number of routers, k

45. Maximum Number of Covered Users • Heterogeneous costs Upper bound Arbitrary solution Greedy: min cost Greedy: min cost w/ specific center Greedy: max coverage Greedy: max coverage w/ specific center Our algorithm Our algorithm w/ specific center Total budget for deployment, B

46. Summary of the simulation results • Our algorithm can be applied to different optimization goals • The ratio between the upper bound and our algorithm matches the approximation ratio • Our algorithms perform better than the greedy heuristics

47. Simulation Results-Use the Census of Taipei

48. Use the Census of Taipei • Use the census to locate the users • Heterogeneous deployment costs: • Higher costs are assigned to locations with higher population density • Goal: Maximize the number of covered users

49. Input 8 km 12 km Total cost of all locations: 60053 Number of users: 7126

50. Output 8 km 12 km The output when the available budget = 15000 Number of covered users: 6600 (≈93% of the total users)