1. Network Design with Concave Cost Functions Kamesh Munagala, Stanford University

2. Single Sink Network Design Given: Source nodes with demands Single sink node Given: Cost on edges Function of demand Design a Network: Connect sources to sink Minimize Cost Sink Sources

3. Concave Cost Functions D(e) = Demand along edge e C(e) = Cost along e C(e) concave function of D(e) Non-decreasing Goal: Min-cost network Theorem: Network is a tree C(e) D(e) 0

4. Why Concave Functions? Economies of Scale More capacity cheaper per unit demand T3 lines cheaper than 40 T1 lines Discreteness in quantity Cannot purchase arbitrarily small capacity Cannot buy 0.1 T1 line

5. Two Problems Cost-Distance Problem: Cost functions could be arbitrary Eg: Multi-level Facility Location [ ACS00,GMM00 ] Our result: O(log n) approximation Edge Installation Problem: Cost function per unit length same on all edges Eg: Buy-at-bulk network design [ SCRS97,AA97 ] Our result: O(1) approximation

6. Previous Results Edge Installation Problem: O(log n log log n) approx [ AA97 ] Based on tree embedding [ Bartal96 ] Special Cases: Steiner Trees: 1.55 approx [ RZ00 ] Maybecast: O(1) approx [ KM00,GMM00 ] Access Network Design: O(1) approx [ GMM00 ] Multi-level FL: O(1) approx [ ACS00,GMM00 ]

7. General Technique Try to aggregate demand: Cheaper to transport Quantify merge cost in terms of OPT Re-solve on the smaller instance OPT on new instance (hopefully) smaller Aggregation Techniques: Cost-Distance: Merge pairs using paths Edge Installation: Merge entire groups using Steiner and Shortest path forests

8. Cost-Distance Problem With: Adam Meyerson, Serge Plotkin Stanford University Appeared in FOCS, 2000

9. Dealing with Concavity Piece-wise linear approximation C(e) D(e) 0

10. The Building Blocks Edge cost: C(e)= F(e) + I(e) D(e) F(e) = Fixed Cost I(e) = Distance Concave functions: Parallel edges with different (F,I) combinations Network has two metrics C(e) D(e)0 F(e) I(e)

11. Fixed Cost Function C(e) = F(e) C(e) independent of D(e) Min-cost network? Steiner Trees C(e) D(e) 0 F(e)

12. Distance Function C(e) = I(e) D(e) C(e) proportional to D(e) Min-cost network? Shortest Paths C(e) D(e) 0 I(e)

13. Formal Statement Given: A set of nodes wishing to communicate to a single sink A multi-graph of feasible edges Each edge has a fixed cost and a distance Select: Edges connecting nodes to sink Minimize: Sum of fixed cost and the shortest- path distances in the respective metrics Assume: Equal demands

14. Numerical Example Sink 0,3 3,1 0,3 3,1 0,3 3,1

15. Solution Sink Cost = 3*3 + 5 + 7 = 21 1 2 1 41

16. Example: Facility Location c(i,j) f(i) d(j) Optimize:  c(i,j) d(j) +  f(i) ( f(i),0 ) Sink ( 0,c(i,j) ) d(j)

17. Solution Idea Merging demand reduces cost: Pair demands using a matching Choose one node in pair as center and send demand to it MC(u,v) = Cost used for matching MC(u,v) = F(u,v) + Dem * I(u,v) Dem = Demand collected at u or v OPT tree has natural matching: Each edge used at most once Cost of matching at most OPT I(u,v)  I(u,s) + I(v,s) Matching in OPT’s solution s uv

18. Algorithm: Basic Iteration Compute min-cost matching Cost at most OPT For each pair Choose random center Send demand there Re-compute on the centers O(log n) iterations Cost of OPT on centers?

19. Optimal Solution Improves Fixed Cost improves E[Distance] cannot worsen Idea: If center closer to sink in OPT, distance improves E[Cost of OPT] improves

20. Main Results Randomized Algorithm E[Cost of Stage]  OPT E[Overall Cost]  O(log n) * OPT De-randomization using LP rounding [ CKN01 ] Only a constant lower bound known 1.48 lower bound for facility location [ GK98 ]

21. Room for Improvement? Intuitively, OPT should be improving Cannot quantify improvement The problem may be too general... What if the cost function is more special? Better merging schemes with quantifiable improvement?

22. Edge Installation Problem With: Sudipto Guha, Univ. of Pennsylvania Adam Meyerson, Stanford University Appeared in FOCS 2000 and STOC 2001

23. Recall... Given: Length metric L(e) on edges Cost per unit length same on all edges C(e) = L(e) * f(D(e)) f(x) concave; independent of edge Aggregation paradigm: Can aggregate multiple demands along forests Contrast with aggregating pairs of demands

24. Motivation Buy at Bulk Network Design [ SCRS97,AA97 ] Provisioning cables by ISPs Cable types: T1: 1.5 Mbps $25/mile T3 : 44 Mbps $400/mile Cost/Length same on all edges! Other examples Steiner Trees Maybecast problem [ KM00,GMM00 ]

25. Simpler Approximation Change of notation: c(e) = Cost/Length along e Same function on all edges Use functions of the form: Fixed Cost: c(e) = F Distance: c(e) = I * D(e) Factor 2 approximation c(e) D(e) 0

26. Fixed Cost Function c(e) = F c(e) independent of D(e) Min-cost network? Steiner Trees c(e) D(e) 0 F

27. Distance Function c(e) = I * D(e) c(e) proportional to D(e) Min-cost network? Shortest Paths c(e) D(e) 0 I

28. c(e) 0 D(e) F U c(e) 0 D(e) F U Max Function I I Basic Building Blocks Min Function Break point U same for all edges!

29. Max: Solution Idea Consolidate along Steiner forest till D(e) > U Then use Shortest paths to sink Shortest Path SteinerD(e) < U D(e) > U c(e) 0 D(e) F U I Steiner Sh. Path

30. Steiner Forest? Construct Steiner tree on demand points and sink Route demands towards sink Cut edges where D(e) > U D(e) > U

31. What’s the Cost? Construct feasible solutions from the optimal solution F(S) 2 OPT OPT pays F on its edges We construct approx Steiner tree I(S) F(S) + OPT Route back fractionally to demands Then route along OPT OPT pays I * D(e) on its edges F(S) + I(S) 5 OPT I(S) OPT F(S) OUR

32. Min: Cost Structure D(e) < U Shortest Path forest D(e) > U Steiner tree to sink c(e) 0 D(e) F U SteinerSh. Path I

33. Shortest Path Forest? Equivalent to the following: Given: A set of demand nodes Length metric on edges Select: A set of facilities Send at least U demand per facility Minimize total routing cost Load Balanced Facility Location O(1) approximation [ KM00,GMM00 ] Demand > U

34. Intuitive Algorithm Construct LBFL solution on demand points Construct Steiner tree to the sink O(1) approximation again [ KM00, GMM00 ] Facilities Demand > U D(e) > U D(e) < U

35. Putting it together c(e) D(e) 0 SteinerSh. Path F I 2F Layered ConstructionPiecewise Approximation D > U1 D > U2 D > U3 U1 U2U3

36. Scaling Technique Make F and I scale: F, 2F, 4F, … I, I/2, I/4, … Lose a constant factor Cost per unit demand reduces dramatically from one layer to next OPT improves significantly on merging demand!!! O(1) approximation c(e) D(e) 0 F > 2F I < I/2

37. Open Problems Better approximation ratio LP rounding - CFL algorithm [ GKKRY01 ] Utilize Steiner nodes better? Multiple sink edge installation O(log n log log n) approx [ AA97 ]