Stateless Optimization of MultiCommodity Flow

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Stateless Optimization of Multi-Commodity Flow

• Baruch Awerbuch
• JHU
• Rohit Khandekar
• IBM Watson

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Main Issue avoiding congestion
Main result Greedy agents operating without
coordination can minimize congestion in
poly-logarithmic time
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Concurrency causes oscillations

• Best response least loaded path

Because of concurrency becomes worst response
Control is needed to avoid oscillations
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Internet perspective

• Since 70s Load-Sensitive routing discarded
• Fixed path routing used
• Routing paths are highly vulnerable to
  DOSattacks masquerading as congestion

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Our framework

• Agents route commodities through a flow-network
  and share network bandwidths
• There is a certain Social objective
• Min the maximum congestion on the links
• Agents are greedy act greedily to minimize their
  own cost no regard to social objective
• Greedy behavior often leads to highly sub-optimal
  performance or even system collapse

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Our approach

• Impose rules of conduct on the agents
• Stateless local rules easy to enforce locally
  without any coordination and without keeping
  track of history
• Induce agents to concurrently converge to a
  near-optimum social objective quickly (typically
  in poly-logarithmic time)

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To Nash

• Traditional approach Analyze Nash equilibrium
• No agent has an incentive to move unilaterally
• Poly-time convergence to Nash via sequential
  moves
• Or, simpler yet, ignore convergence issue all
  together
• Does this make sense in a distributed and dynamic
  system?
• System is distributed agents dont move
  sequentially
• In poly-time system changes thus no convergence

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or Not To Nash?

• We define a notion of aggregate equilibrium.
• Where system state does not change by too much in
  long-enough period of time
• Aggregate equilibrium implies near-optimality.
• While not in aggregate equilibrium
• Irreversible significant progress
• Eventually in Aggregate equilibrium.

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Concurrent Multi-commodity Flows

• a graph G(V,E,C) edge-capacities c(e)
• k commodities
• source si, sink ti, demand di 0

For each commodity route di flow between si and
ti such that the maximum edge congestion is
minimized.
?i fi(e)
f(e)
total flow thru e
congestion(e)


u(e)
u(e)
capacity of e
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Concurrent Multi-commodity Flows
ce capacity
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Concurrent Multi-commodity Flows
d(5)
d(2)
d(1)
d(4)
d(3)
Route all demands and minimize the max
edge-congestion.
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Previous sequential solution

• Many combinatorial algorithms known
• Shahrokhi-Matula (1990)
• Klein-Plotkin-Stein-Tardos (1990)
• Leighton-Makedon-Plotkin-Stein-Tardos-Tragoudas
  (1991)
• Plotkin-Shmoys-Tardos (1991)
• Garg-Könemann (1998)
• Fleischer (2000)
• Young (2001)

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Previous Work

• Even-Dar and Mansour 05 complete network
• symmetric strategy space
• Fisher, Räcke, Vöcking 06 another congestion
  model
• Infinitely many agents each controlling
  infinitesimal flow.
• Single commodity (symmetric strategy space).
• Fisher Vöcking (2004) , Chien Sinclair
  (2007)
• Sequential games
• polynomial convergence to Nash equilibrium

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Stateless algorithms

• Algorithms reacting to the current state of the
  system without keeping history
• Output function (State)
• Greedy algorithms are a special case of
  stateless algorithms

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Properties of stateless algs

• Incremental operation we do not start from
  scratch upon each change
• Self-stabilization system corrects itself
  after transient failures
• There is no need to initialize consistently

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Components of our framework

• Load-sensitive pricing of the edges
• flow agents are forces to pay these prices
• Flow control (speed limit) rule
• cannot increase or drop the flow too fast
• Profit margin (inertia) rule
• rerouting must yield ? gt 0 profit margin

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Opportunity cost

• Cost of an edge with flow f (m1/e)f(e)

Opportunity cost
congestion
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Algorithmic Framework

• We want to minimize the maximum flow through any
  edge
• minimize maxe f(e)

We use a smooth convex equivalent
function minimize ? ?e (m1/e)f(e)
Fact mO(1)-approx. of ? implies (1O(e))-approx.
of maximum congestion
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Concurrent Algorithmic Framework

• Maintain the correct estimate of the derivative
• During the flow rerouting, the lengths l(e)
  should not change by more than a factor of (1e).
• ?l(e) l(e) log (m1/e) ?f(e)
• l(e) e
• ?f(e)

e2
log m
Flow control constraint
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Flow control for concurrency

• A flow cant increase by more than 1?
• A flow cant decrease by more than 1-?-
• ?- L ? , i.e., downwards speed limit is
  more aggressive than upwards limit
• Agents are forced to obey the speed limits

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Effect of speed limit
Flow (log scale)

• Fast increase, slow decrease

time
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Inertia rule

• Profit margin (inertia) rule
• rerouting must yield ? gt 0 profit margin

1 ?
b
a
d
c
1
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Algorithm run by each flow

• Graph residual capacity speed limits
• while
• non-saturated path exists at a cost of (1-?)
  below the average cost, and
• Less than 1-?- fraction of demand rerouted
• Saturate this path, by increasing its flow to
  1? times the flow on the bottleneck edge
• Compensate by proportional uniform decrease

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Blocking Flow along Shortest Paths
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Blocking Flow along Shortest Paths
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Blocking Flow along Shortest Paths
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Blocking Flow along Shortest Paths
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Summary Bounded Best Response Dynamics

• We impose congestion-sensitive (exponential)
  edge-costs.
• Each agent reroutes its flow to minimize its own
  cost subject to
• flow control rule cant ramp up too fast
• inertia rule dont bother with minor
  improvements
• Does this bounded best response dynamics converge
  to a near-optimum solution?
• If yes, how fast?

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Main idea of proof

• We define the notion of aggregate equilibrium
  (weaker than Nash)
• We show that aggregate equilibrium yield
  near-optimality
• We show that non-equilibrium state will
  eventually involve large improvement in a
  potential function

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Showing potential decrease

• Without speed limits, it would be easier to claim
  potential improvement in moving from expensive to
  cheap routes
• We show that speed limit achieves the same, in
  spite of ghost chasing problem, namely shortest
  path changing very frequently.

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Main Result
Starting from an arbitrary flow, the flow
converges to a 1? approximation to the minimum
max-congestion in of rounds upper bounded
by Here m edges, P paths C
maxj Cj/minj Cj
Self-stabilizing
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Conclusion

• These ideas can be extended to other packing and
  flow problems.
• Open question Eliminate the dependency on L in
  the convergence time and get a completely
  poly-logarithmic convergence?