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Stateless Optimization of Multi-Commodity Flow
Baruch AwerbuchJHU
Rohit KhandekarIBM 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 70’s: 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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• 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 don’t move sequentially– In poly-time system changes; thus no convergence
To Nash …
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• 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.
… or Not To Nash?
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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.
f(e)congestion(e)=
total flow thru e
capacity of e=
u(e)
∑i fi(e)=
u(e)
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Concurrent Multi-commodity Flows
ce = capacity
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Concurrent Multi-commodity Flows
d(1)
d(5)
d(2)
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 alg’s
• 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 profit margin
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Opportunity cost
• Cost of an edge with flow f = (m1/ε)f(e)
congestion
Opportunity cost
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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 ф = ∑e (m1/ε)f(e)
Fact: mO(1)-approx. of ф implies (1+O(ε))-approx. of maximum congestion
ф
f
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• Maintain the correct estimate of the derivative:
During the flow rerouting, the lengths l(e) should not change by more than a factor of (1+ε).
Δl(e) = l(e) · log (m1/ε) · Δf(e)
≤ l(e) · ε
Δf(e) ≤ log m
Concurrent Algorithmic Framework
ε2
Flow control constraint
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Flow control for concurrency
• A flow can’t increase by more than 1++
• A flow can’t 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
• Fast increase, slow decrease
time
Flow(log scale)
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Inertia rule
• Profit margin (inertia) rule:– rerouting must yield profit margin
a
c
b
d
1+
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: can’t ramp up too fast– inertia rule: don’t 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?