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Algorithms for Concave Cost Network Flow Problems

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Algorithms for Concave Cost Network Flow Problems

Kamesh Munagala

Stanford University

- 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

Cost Structures in Network Design

- 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

Cost

Fixed

Cost

Storage Capacity

- 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

Cost

One

Truck

Load Transported

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

- 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

Cost

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

0

Demand

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

Warehouse cost = f(i)

Warehouse i

Transportation Cost = c(i,j)

Outlet j

Demand d(j)

Optimize:c(i,j) d(j) + f(i)

Sink

f(i)

i

c(i,j)

d(j)

Optimize:c(i,j) d(j) + f(i)

Sink

- 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

- 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

- Provably close to optimal solution on all instances

Cost

1

0

Flow

Sink

1

3

2

Flow = 1

1

Cost = 5

- Operations Research:
- Uncapacitated Fixed Charge Problem
- Magnanti, Mireault, Wong. 1986
- Hochbaum, Segev. 1989
- Ortega, Wolsey. 2000

- No approximation algorithms known for this problem

- Logarithmic approximation
- [Meyerson, Munagala, Plotkin. 2000]

- Simple to implement
- Uses shortest path and greedy matching computations

- Efficient in practice
- Approximation ratio much better on real data

- Best approximation result till date
- De-randomization [Chekuri, Khanna, Naor. 2001]
- Best hardness: 1.47 [Guha, Khuller. 1998]

- 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

s

u

v

- 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

- 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

Special Cases of Concave Cost Flow

Warehouse cost = f(i)

Warehouse i

Transportation Cost = c(i,j)

Outlet j

Demand d(j)

Optimize:c(i,j) d(j) + f(i)

- 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

- 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

Production Units

g(k)

c(k,i)

Warehouses

f(i)

c(i,j)

Outlets

d(j)

2-level Warehouse Location

- 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

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

- 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

- 3.27 approximation

- 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

- Cable types:
- 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]

- 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

- If dist(a,b) < dist(a,c)
- General concave cost flow:
- Closeness is a function of demand transported

- 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]

- Partitions have smaller diameter than original graph

Diameter of Graph = D

Diameter < D/2

w.h.p. Distances > D/log n

Route from centers of children to center of parent

- 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

- 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

- 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

Partition assuming T3 line becomes cheaper at 10 Mbps bandwidth

Aggregation point

Groups with 10 Mbps total bandwidth

T1 lines

T3 lines

- 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

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

- 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]

- 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

- 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?

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

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