1. On collection depots location problems
Binay K Bhattacharya
School of Computing Science
Simon Fraser University
(Joint work with Robert Benkoczi, Sandip Das, Jeff Sember and Arie Tamir)

2. What is facility locating?
Facility provides some kind of service.
Client needs this kind of service.
General office, information center
‘GOOD’
SERVICE

3. Tower may be positioned anywhere – (continuous)
Where should the radio tower be located?
Tower may be positioned anywhere – (continuous)
Tower may be positioned in five available slots – (discrete)
Tower may be located on the roadside – (network)

4. Distance metric Minkowski distance (Euclidean, Manhattan)
Network distance

5. Classical k-center problem
Given a universe U (where facilities can be located), a set of clients C, a metric d, and a positive integer k, a k-center of C is a set of k points F of the universe U that minimizes
maxjεC {miniεF dij}.

6. Classical k-median problem
Given a universe U, a set of points C, a metric d, and a positive integer k, a k-median of C is a set of k points F of the universe U that minimizes
ΣjεC {miniεF dij}.

7. Euclidean k-center Minimize the maximum distance to the facility

8. Euclidean k-median Minimize the average distance to the facility

9. Euclidean k-center
(Megiddo and Supowit, 1984): k-center is NP-hard.
(Feder and Green 1988): ε-approximation remains NP-hard for any ε < (1+√7)/2 ≈
Single facility location problem
Megiddo (1983) proposed a linear time algorithm.

10. Euclidean k-median
Meggido and Supowit (1984): Problem is NP-hard in the plane, and cannot be approximated to within 3/2.
Varignon frame (Classical MinSum)

11. Euclidean k-median
Meggido and Supowit (1984): Problem is NP-hard in the plane, and cannot be approximated to within 3/2.
Bajaj (1986) showed that the problem cannot be solved exactly in the plane using radicals.
Numerical methods are efficient.
Approximation scheme using ellipsoid method proposed by Chandrasekhar and Tamir (1990)

12. Settings Client (demand service) Facility (service center)
Collection Depots

13. Express Transportation
Application (1)
Express Transportation

14. Application (2)
Garbage collection

15. Survey of collection depots problem
Drezner and Weslowsky (2001)
- on the line, an optimal MinSum solution must exist either on a customer point or a collection depot.

16. Survey of collection depots problem
Drezner and Weslowsky (2001)
- on the line, an optimal MinSum solution must exist either on a customer point or a collection depot.
- on a line, an optimal solution utilizes at most two depots.

17. Survey of collection depots problem
Tamir and Halman (2005) studied the multi-facility MinMax and MinSum depots problems in the plane, in graphs, in trees and in paths.
- introduced three versions of the depots problem:
each customer is given a set of depots that the customer is allowed to use.
customer one-way: facility  depot  customer
depot one-way: facility  customer  depot

18. Survey of collection depots problem
Tamir and Halman (2005):
- 3-approximation solution for the k-depots MinMax problem in general graphs.
- O(p+n2lgn) optimal solution for the k-depots MinMax problem in trees.
- For 1-depot MinMax problem
paths: O(n+p)
trees: O(p+n lgn)
plane: O(p2n2 lg3(pn)) (Euclidean)
O(pn lg4(pn) (Manhattan)

19. Survey of collection depots problem
Benkoczi, Bhattacharya and Tamir (2007) considerd
MinMax and MinSum depots problems in trees.
- 1-depot MinSum : O((n+p)lg(n+p))
- 1-depot MinMax : O(n+p)
- k-depot MinSum : O(p+kn3)

20. Survey of collection depots problem
Benkoczi, Bhattacharya, Das, and Sember (2005, 2008) considerd MinMax and MinSum depots problems in the plane.

21. Outline of the rest of the talk
Our results (MinMax and MinSum problems in the plane)
-Voronoi diagrams
-Feasible assignments
-Approximation algorithms
-The Barrier problem
-The Room problem

22. Euclidean collection depots problem

23. Voronoi diagrams

24. Voronoi diagrams

25. Voronoi diagrams (Edges)

26. Voronoi diagrams (Properties)

27. Voronoi diagrams (Properties)

28. Voronoi diagrams (Constructions)

29. Voronoi diagrams (Constructions)

30. Voronoi diagrams (Constructions)

31. Voronoi diagrams (Constructions)

32. Voronoi diagrams (Constructions)

33. Voronoi diagrams (Constructions)

34. Voronoi diagrams (Constructions)

35. Voronoi diagrams (Constructions)

36. Voronoi diagrams (Constructions)

37. Voronoi diagrams (Constructions)

38. Voronoi diagrams (Constructions)

39. Voronoi diagrams (Constructions)

40. Voronoi diagrams (Constructions)

41. Feasible assignments

42. Feasible assignments

43. Merged Voronoi diagrams of customers

53. MinSum depots problem

54. MinSum depots problem *

55. MinMax depots problem
MinMax depots problem can be solved in O(p2n2) times the time it takes to solve Classical MinMax for O(n) points.
Classical MinMax problem can be solved in O(n) time (Megiddo(1983))
Feasibility test of the MinMax depots problem can be solved in O(n2p2lg(pn)) time (Tamir and Halman(2005)). This results in O(n2p2lg3(pn)) solution for the general problem.

56. Barrier problem

57. Room problem

58. Classical MinMax problem

59. MinMax single collection depot problem