Another story on Multi-commodity Flows and its

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Another story onMulti-commodity Flowsand its
dual Network Monitoring

• Rohit Khandekar
• IBM Watson
• Joint work with
• Baruch Awerbuch
• JHU

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Outline

• Crash course
• Set cover problem and the greedy algorithm
• Framework for distributed covering problems
• The maximum multi-commodity problem and its dual
  passive commodity monitoring problem
• Fast converging distributed approximation schemes

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The Set Cover Problem

• Given
• a set of elements U
• subsets S1, S2, , Sk µ U with costs c1, c2, ,
  ck 0
• Find
• Minimum cost collection of subsets whose union is
  entire U.

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The Greedy Algorithm
(re 1 if e is not yet covered)
Gives O(log n) approximation where n U.
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The Fractional Set Cover Problem
The LP relaxation of the set cover IP.
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The Fractional Greedy Algorithm
Drawback iterations n/²2
Gives O(log n) ( 1 ² ) approximation.
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The Fractional Greedy Algorithm
all
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The Fractional Distributed Algorithm
Also computes a near-optimum dual solution
Luby-Nissan (93), Garg-Konemann (98), Young (01)
iterations
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Concurrent Multi-commodity Flow
Maximum Throughput
ce capacity
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Concurrent Multi-commodity Flow
Maximum Throughput
Send maximum total flow between the pairs subject
to the edge-capacity constraints.
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Concurrent Multi-commodity Flow
Maximum Throughput
Send maximum total flow between the pairs subject
to the edge-capacity constraints.
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Distributed Computation Model

• The ROUTERS model
• Intelligence is embodied in the network routers
• Computations takes place by exchanging messages
  between neighboring routers
• Complexity measures
• Approximation ratio ((1²) approximation)
• Message congestion ( messages/router/round)
• Space complexity (space needed/router)
• Convergence time ( rounds to converge)
• Computational complexity (total work)

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Multicommodity Problem Its Dual
dual set cover edges sets paths elements
Dual Probe edges e with frequency xe so that
each path gets probed to an extent 1 while
minimizing the total cost of probing ?e ce xe
Passive commodity monitoring
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Main Result

• There is an algorithm for maximum multicommodity
  flows and passive commodity monitoring with the
  following properties
• approximation
• 
  convergence
• space and
  messages/router
• computational overhead

L maximum hop-length of a flowpath
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Comparison with Previous Work
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The Algorithm

• Set cover with edges as sets and paths as
  elements
• Associate with each path p, a residual
  requirement
• (profit of path p)
• ( is a constant)

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The Algorithm

• Repeat
• For all edges that (approximately) minimize the
  cost-to-profit ratio
• increase
• Increase the flow on all paths through such edges

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How to compute aaaaaaaa
Compute
A shortest path algorithm (Dijkstra) computes
A similar (dynamic programming) algorithm
computes
Computing shortest paths on a semi-ring
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How to compute aaaaaaaa
?1
l1
?2
l2
?3
l3
l4
?4
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Conclusions

• First multi-commodity algorithm
• Via dual multi-cut problem
• Breaks the ?(m) convergence barrier
• Convergence polynomial in path-length L
• Question Can we get O(L) convergence?

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Thank You