Network Design Adam Meyerson Carnegie-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 $

Concave Cost Functions $ 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 $ 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 $/m

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$/m

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 $/m

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?