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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?