Algorithms for Concave Cost Network Flow Problems

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Algorithms for Concave Cost Network Flow Problems

• Kamesh Munagala
• Stanford University

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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

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Cost Structures in Network Design

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Warehouse Location
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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

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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

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Transportation Cost

• Linear in distance to outlet
• Linear in load transported to outlet
• Minimum fixed cost for one truck

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Features of Cost Structure

• Economies of Scale
• More capacity cheaper per unit demand
• Applies to warehouse costs
• Discreteness in 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

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Modeling Allocation Costs
Cost

• Cost is
• non-decreasing
• concave
• function of demand serviced

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Demand
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Concave Cost Flow Problem

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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
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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)
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Modeling Facility Location
Sink
f(i)
i
c(i,j)
d(j)
Optimize ? c(i,j) d(j) ? f(i)
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Solution
Sink
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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

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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
3
1
2
Flow 1
1
Cost 5
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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

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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

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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

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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 OPTs solution
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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

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Special Cases of Concave Cost Flow

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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)
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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

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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

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

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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
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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

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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

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Why is this problem simpler?

• Notion of close-by
• If dist(a,b) lt 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

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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

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Partitioning
Diameter of Graph D
Diameter lt D/2
w.h.p. Distances gt D/log n
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Routing
Route from centers of children to center of parent
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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

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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

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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 gt 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-up instead of
  top-down
• Properties of partition
• Close-by demands still grouped together
• The cost function decides group boundaries

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First Aggregation Step
Partition assuming T3 line becomes cheaper at 10
Mbps bandwidth
Aggregation point
Groups with 10 Mbps total bandwidth
T1 lines
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Complete Solution
T3 lines
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Constructing the Partitions

• Given
• A set of demand nodes
• Length metric on edges
• Select Set of aggregation points
• Send at least U demand per point
• Route along shortest paths
• Minimize total routing cost
• Load Balanced Facility Location
• O(1) approximation KM00,GMM00
• Iteratively construct larger partitions

Demand gt U
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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
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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

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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

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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?

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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

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Acknowledgements

• Theory wing
• Mayur Datar, Aris Gionis, Gagan Aggarwal, Keyvan
  Mohajer, Liadan OCallaghan, 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