Algorithms for Concave Cost Network Flow Problems

1. Algorithms for Concave Cost Network Flow Problems Kamesh Munagala Stanford University

2. Talk Outline • Motivation via simple example • Concave cost flow problem: • Formal problem statement • Simple Randomized Algorithm • Special Cases: • Motivation from networking problems • Our results • The buy-at-bulk algorithm

3. Cost Structures in Network Design

4. Warehouse Location

5. Decision Problem • Costs: • Opening and operating warehouse • Shipping demand • Tradeoff: Lots of warehouses implies low shipping cost • Optimize: Linear combination of costs • Decisions: • How many warehouses to open • Where to open warehouses • How to ship to outlets

6. Cost Fixed Cost Storage Capacity Warehouse Cost • Minimum fixed cost for operating warehouse • Additional cost depending on storage capacity needed • Typically reduces as capacity increases • Example: Staff does not double with doubling capacity

7. Cost One Truck Load Transported Transportation Cost • Linear in distance to outlet • Linear in load transported to outlet • Minimum fixed cost for one truck

8. Features of Cost Structure • Economies of Scale • More capacity cheaper per unit demand • Applies towarehouse costs • Discretenessin quantity • Cannot purchase arbitrarily small capacity • Applies to warehouse and transportation costs • General phenomena in network design: • Costs of caches, routers and cables obey these properties

9. Modeling Allocation Costs Cost • Cost is: • non-decreasing • concave function of demand serviced 0 Demand

10. Concave Cost Flow Problem

11. Concave Cost Flow Problem • Given: • Undirected network • Cost on edges • Concave function of demand • Many demand nodes • Distinguished sink node • Compute: • Minimum cost flow Sink Sources

12. Facility Location Warehouse cost = f(i) Warehouse i Transportation Cost = c(i,j) Outlet j Demand d(j) Optimize:c(i,j) d(j) + f(i)

13. Modeling Facility Location Sink f(i) i c(i,j) d(j) Optimize:c(i,j) d(j) + f(i)

14. Solution Sink

15. Other Special Cases • Steiner Trees • Probabilistic Steiner Trees [KM00] • Multilevel Facility Location • Buy at Bulk Network Design [SCRS97] • Applications in network design: • Multicast tree design • Hierarchical placement of caches and routers • Placement of web content in caches • Buying cables to provision bandwidth

16. Hardness of the Flow Problem • Facility location is NP-Hard • Steiner Tree Problem: • Fixed cost for using edge • NP-Hard [Karp. 1972] • Approximation algorithms: • Provably close to optimal solution on all instances • Example: Cost  5 OPT • Polynomial running time Cost 1 0 Flow Sink 1 3 2 Flow = 1 1 Cost = 5

17. Previous Results • Operations Research: • Uncapacitated Fixed Charge Problem • Magnanti, Mireault, Wong. 1986 • Hochbaum, Segev. 1989 • Ortega, Wolsey. 2000 • No approximation algorithms known for this problem

18. Our Result • Logarithmic approximation • [Meyerson, Munagala, Plotkin. 2000] • Properties of our algorithm: • Simple to implement • Uses shortest path and greedy matching computations • Efficient in practice • Approximation ratio much better on real data • Subsequent Results: • Best approximation result till date • De-randomization [Chekuri, Khanna, Naor. 2001] • Best hardness: 1.47 [Guha, Khuller. 1998]

19. Basic Algorithm • Merging demand reduces cost: • For every pair (u,v) compute min cost path in graph to send demand from u to v or vice versa • Let this be cost of (u,v) edge • Compute min cost matching in this complete graph • Pair demands using this matching • Choose one node in pair as center and send demand to it • Number of demand nodes halves • Repeat logarithmic times

20. s u v Proof Idea • The optimal solution encodes a matching of nodes • Implies cost of matching at most cost of optimal solution • [Marathe et al 1998] Matching in OPT’s solution

21. Problem too hard? • Which node is cheaper to route to depends on demand being routed • Hard to make decisions about merging a whole group of nodes • Not enough structure in solution • Except for the fact that it encodes a matching • Best hardness result known is only 1.47 • Guha, Khuller. 1998

22. Special Cases of Concave Cost Flow

23. Facility Location Warehouse cost = f(i) Warehouse i Transportation Cost = c(i,j) Outlet j Demand d(j) Optimize:c(i,j) d(j) + f(i)

24. Previous Results • Operations Research: • Kuehn, Hamburger. 1963 • Cornuejols, Fisher, Nemhauser. 1977 • Approximation Algorithms: • Guha, Khuller. 1998 (Lower bound = 1.47) • Mahdian, Ye, Zhang. 2002 (1.52 approx) • Fast combinatorial algorithms known [CG99,JV99,AGKMMP01] • Applications: • Centroid based clustering • Placement of caches and replicated data objects: • Minimize latency of user access

25. Our Result • Novel variant of facility location: • Each facility needs to satisfy minimum amount of demand • Load Balanced facility location • Constant factor approximation algorithm [KM00,GMM00] • Reduction to classical facility location • Applications: • Subroutine in concave cost flow algorithms • Solving clustering variants [GM02] • Favor either large or small cluster sizes

26. Multilevel Facility Location Production Units g(k) c(k,i) Warehouses f(i) c(i,j) Outlets d(j) 2-level Warehouse Location

27. Previous Results • Problem formulation: • Kaufman, vanden Eede. Hansen, 1977 • Factor 3 approximation: • Aardal, Chudak, Shmoys. 1999 • Exponential size linear program • Can be solved using Ellipsoid algorithm • Very inefficient in practice • Application in networks: • Hierarchical placement of caches, switches and routers

28. Modeling as a Flow Problem Two copies of the network Outlets f(i) i i g(k) Sink k c(i,j) c(i,j) Route flow from outlets to the sink node

29. Our Results • Simple combinatorial algorithm: • 9 approximation [GMM00] • Reduce to classical facility location • Can now use very efficient algorithms • Subsequent results: • 3.27 approximation • Ageev, Ye, Zhang. 2002 • Combinatorial algorithm

30. Buy-at-bulk Network Design • Provisioning cables to route data to core network • Bandwidth cost obey economies of scale • Cable types: • T1: 1.5 Mbps $30/mile $20/Mbps/mile • T3: 44 Mbps $440/mile $10/Mbps/mile • Cost of cables is a concave function • Metrical special case: • Cost of bandwidth same per unit length everywhere • Concave function same per unit length on all edges • [Salman, Cheriyan, Ravi, Subramanian. 1997]

31. Why is this problem simpler? • Notion of close-by: • If dist(a,b) < dist(a,c) • Cheaper to transport demand from a to b than to c • Independent of demand transported • Natural algorithm: • Merge close-by demands together • Cheaper to transport this merged demand to a far away place • General concave cost flow: • Closeness is a function of demand transported

32. Recursive Metric Partitioning • Just focus on the metric space • Ignore the cost function completely • Recursively partition graph based on closeness (randomized): • Partitions have smaller diameter than original graph • [Bartal96, Bartal98, CCGG98, CCGGP98] • Nodes in different partitions far away from each other w.h.p. • For each partition, have a center node • Collect all demand within a partition at center node • Send this demand to the center of the parent of this partition • [Awerbuch, Azar. 1997]

33. Partitioning Diameter of Graph = D Diameter < D/2 w.h.p. Distances > D/log n

34. Routing Route from centers of children to center of parent

35. Discussion • Paradigm of aggregation: • Group together close-by demand nodes • Reduce cost of transportation • Problems with approach: • Same partition for all cost functions • Some close-by nodes bound to end up in different partitions • Problem even if graph is just a cycle • Worst case logarithmic performance expected in practice

36. Other Approaches • Linear Programming: • Andrews and Zhang. 1998 • Improve the logarithmic ratio for special cases • Usually produces optimal integer solutions in practice • The size of the program is huge: • N3 variables • Inefficient in practice • Simple algorithms known for very special cases: • Salman, Cheriyan, Ravi, Subramanian. 1997

37. Our Solution Idea • Use cost function to construct the partitioning: • Say we have T1 and T3 lines • Say cheaper to use T3 line if bandwidth > 10Mbps • Then, we should find: • Min cost way of aggregating demands using T1 lines • Each aggregated node receives 10Mbps bandwidth • Min cost way of connecting aggregated nodes to sink node • Construct partitioning bottom-upinstead oftop-down • Properties of partition: • Close-by demands still grouped together • The cost function decides group boundaries

38. First Aggregation Step Partition assuming T3 line becomes cheaper at 10 Mbps bandwidth Aggregation point Groups with 10 Mbps total bandwidth T1 lines

39. Complete Solution T3 lines

40. Constructing the Partitions • Given: • A set of demand nodes • Length metric on edges • Select: Set of aggregation points • Send at leastUdemand per point • Route along shortest paths • Minimize total routing cost • Load Balanced Facility Location • O(1) approximation [KM00,GMM00] • Iteratively construct larger partitions Demand > U

41. One Issue Routing with a cable type need not be along shortest paths Capacity = 1 Cost/Length = 1 1 1 0.5 Case 1: Cost = 1.5 Cost = 2 Demand = 0.5 Case 2: Cost = 2.5 Cost = 2 Demand = 1.0

42. Another Issue • We are constructing partition bottom-up • Optimal partition could look different • If we make error in first grouping, error propagates upward • How do we bound cost against optimal cost • Scaling technique: • Observation: Error propagates only if similar cable types exist • Eliminate all cable types that look similar except one • Partitioning at every stage close to optimal partitioning • Constant factor approximation [GMM00,GMM01]

43. Properties of Algorithm • Simple to implement: • Uses facility location and Steiner trees as subroutines • Very efficient in practice • Preliminary experimental results: • Real ISP and geographic data • Real cable types and costs • At most 10% away from optimal solution • Subsequent work: • Talwar. 2002 (213 approx) • Gupta, Kumar, Roughgarden. 2003 (72 approx) • Based on the ideas in our algorithm

44. Open Problems • Better approximation ratios: • Buy-at-bulk: 72 [GKR03] • Concave cost flow: Logarithmic approximation [MMP00] • Multiple sink concave cost flow: • Aggregation paradigm fails! • Buy-at-bulk problem: • Logarithmic approximation [AA97] • Aggregation paradigm applicable to other problems?

45. Acknowledgements Research collaborators: • Serge Plotkin, Stanford University • Abhiram Ranade, IIT Bombay • Sudipto Guha and Adam Meyerson • Matthew Andrews, Bell Laboratories • Pat Brown, Stanford University School of Medicine • Ramesh Hariharan, Strand Genomics Pvt. Ltd. • Zoe Abrams, Ashish Goel, Baruch Schieber, Debasis Mitra, Devavrat Shah, Jochen Konemann, Maxim Sviridenko, Rina Panigrahy, Rob Tibshirani, Shankar Krishnan, Suresh Venkat and Tracy Kimbrel

46. Acknowledgements Theory wing: • Mayur Datar, Aris Gionis, Gagan Aggarwal, Keyvan Mohajer, Liadan O’Callaghan, Majid Emami, Moses Charikar and Piotr Indyk Friends : • Dhananjay Gore, Rohit Nabar, Aditi Nabar, Kumar Muthuraman, Mohan Lakhamraju, Nandan Das, Prashanth Hande and Sameer Siruguri Parents and Roopa