Network Design

Adam MeyersonCarnegie-Mellon University

Multi-commodity Flow

• Given Graph G=(V,E)• Send s-t flows• Many s-t pairs

• Pay cost to send flow over an edge

• Minimize total cost

Cost Function on an Edge

• Pay per unit flow

• Until capacity

• Exact curve depends on the edge

• Goal: Separate Paths

Flow

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Concave Cost Functions

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Flow

Why Concave Costs?

• “Buy-at-Bulk” discounts, for example:– Product Transport– Network Bandwidth– Device Purchase

• Goal: Merge Paths!

Open Problem

• Given graph, concave costs, s-t pairs

• Construct multi-commodity s-t flows

• Minimize total cost

• NP-Hard! No non-trivial approx. known!

What’s known?

• Single sink (symmetric sink)– O(log n) approximation– MMP00, CKN00

• Single function (with scaling)– O(log n log log n) approximation– AA97 w/ B98, CCGGP98

• Single sink and single function– O(1) approximation – GMM01

General Concepts

• Good to merge demand

• Better large demand in few places

• Must choose “good” merge points

Model

• Piecewise linear

• Fixed/Incremental

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Flow

Special Cases

• Fixed Cost only: Steiner Tree/Forest

• Incremental Cost only: Shortest Paths

Single Sink

• Pair up demands• Random Merge• Each step < OPT• E[OPT] decreases• O(log n) approx.• Fast algorithm using

approximate matching

De-randomizing

• Solve linear program at each step

• Use LP to determine the merging

• Again O(log n) approximation

• Now bounded against the LP fractional

• Running time maybe not so fast

Open Problems

• What is the LP-IP gap of CKN program?– At least 2, at most O(log n)– Exact gap is unknown!

• Is O(1) approximation possible here?

Single Function

• Distance / scaling• What if on a tree?

– Easy to route, merge!

• Metric is general

Flow

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Bartal’s Results

• Embed any metric into probability distribution on trees

• E[t(x,y)] < O(log n log log n) d(x,y)

• Hardness of Omega(log n) for this

Algorithm for Single Function

• Embed distance into tree via Bartal

• Solve aggregations on tree

• Map back into original metric space

• O(log n log log n) approximation

Open Problems

• Is tree embedding the best approach?

• Is O(log n) best-possible for this problem?

Single Sink and Function

Flow

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Why is this easier?

• Always good to merge demand

• We know how much to merge at each step

• Still must find locations for merging

Flow

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Facility Location Problem

• Locate facilities such that:– Small average demand-facility distance

• Each facility must serve large demand– Otherwise, facilities come “for free”

• Load Balanced Facility Location (LBFL)

LBFL Algorithm

• Incremental Costs dominate

• Compute cost to gather requirement

• Solve FL with costs

• Bicriteria approx.

Steiner Forest Problem

• Given demand nodes• Fixed cost dominates• Construct forest

• Each tree contains at least D demand

• Open!

Single Sink+Function:

• Build alternate LBFL, Steiner trees

• Amortize costs separately for steps

• Bound Steiner steps against “one big tree”

• Double geometric sum, O(1) but large

Open Problems

• O(1) for LBFL without violating bounds

• O(1) for Steiner forest problem

• Better constant approximations

Online Network Design

• Demand points arrive one at a time

• General case as hard as online Steiner tree

• Tree embedding algorithm is online

Access Network Online

• Special case of single sink and function

• Simple algorithm: choose cable randomly

• O(1) v.s. random order inputs

• O(log k) v.s. adversarial order inputs

• Can we do better? Better analysis?

• Does something like this work w/o access network assumptions to give O(log n)?

More Open Questions

• What about arbitrary functions?– For example, concave capacitated

• What about redundancy?