A General Approach to Online Network

Optimization Problems

Seffi NaorComputer Science Dept.

TechnionHaifa, Israel

Joint work: Noga Alon, Yossi Azar, Baruch Awerbuch, and Niv Buchbinder

The Set Cover ProblemInput:• X = 1, 2, ... ,n – ground set of

elements.• S – family of subsets of X.• c – cost function on S.Goal:A min cost collection of sets from

S that cover X.Classic: greedy algorithm is an

O(logn)-approximation.

The Online Set Cover Problem

• An adversary gives the elements one-by-one to the algorithm.

• When a new element arrives, the algorithm must cover it by a set from S.

• X’ – Elements given by adversary ( ).

Competitive Factor:

'X X

Cost of sets used by online algorithm

Cost of optimal solution of X'

Example (1)

• The sets are servers and the elements are potential clients.

• Each server can provide the service to a subset of the clients.

• There is a setup cost for activating a server.

• Clients arrive one-by-one.

Example (2)

Input:• X = {1,2, … ,n} – a ground set.• S – All subsets of X of size .Game:• Adversary gives uncovered element at each step.• Online algorithm picks a set.

Termination: All elements are covered:

Performance:• Competitive ratio is at least

n

n

n

| ' |X n

n

Example (2) (contd.)

Good news or bad news?

Not so bad …

Competitive Ratio is O(log m).

Depends on both n and m (unlike offline case).

log logn

n nn

Graphical Representation

r

ea

50 100 150

bab c ed c

Request for element a: Purchase a path from r to a leaf labeled a.

Network Optimization Problems

Network = Weighted graph, directed or undirected

Demands: Disjoint sets of vertices Di = (Si, Ti)

Problems:

Connectivity - Connect the sets by “picking” edges such that there is path from a vertex in Si to a vertex in Ti.

Network Optimization Problems (contd.)

Problems (contd.):

Cuts - Disconnect the sets by “removing” edges such that each vertex in Si is disconnected from each vertex in Ti.

Goal:

Minimize the total cost of picked or removed edges.

Online Network Optimization Problems

• Network and weight function are known in advance to the online algorithm.

• The demands Di = (Si, Ti) are given one-by-one. Each demand is satisfied upon arrival by purchasing edges.

• Competitive factor is ratio between: cost of edges purchased by the online algorithm and cost of optimal solution.

Connectivity Problems - Examples

Online (Non-Metric) Facility Location:• There are potential locations of facilities.• Each location has a “setup cost”.• Clients arrive one-by-one.• Each client may connect to each facility by

paying a “connection cost”.

Goal:• Decide which facilities to open to minimize

the total cost:• “Total Opening Cost” + “Total Connection

Cost”

Connectivity Problems - Examples

Online Multicast Problem:

• A family of arbitrary rooted trees, where the tree edges have costs.

• Each tree leaf is associated with a subset of the clients.

• Clients arrive one-by-one.• Upon arrival of a client: a path from a

leaf (associated with the client) to a root has to be purchased.

Goal: Minimize cost of purchased edges.

The Multicast Problem

Connectivity Problems - Examples

Online Group Steiner problem in trees:

• Same as the multicast problem – but now there is a single arbitrary rooted tree.

• This means that paths from leaves associated with the same client to the root are not necessarily disjoint.

Goal: Minimize total cost of purchased edges.

Online Group Steiner in Trees Problem

Cuts Problem - Example

Online Multicut Problem:

• General weighted undirected Graph• Demands: pairs of vertices Di = (si, ti)

Goal:• Disconnect each pair Di = (si, ti) by

removing edges from the graph.

• Minimize the total cost of edges.

Online Multi-cut Problem

S3

T3

S1 T1

S2

T2

Fractional Network Problems

For each demand (S,T): Connectivity Problems: • Give fractional weights to edges s. t.

maximum flow from S to T is at least 1.

• Minimize c(e) w(e) Cut Problems: • Give fractional weights to edges s. t.

distance from S from T (closest vertices) is at least 1.

• Minimize c(e) w(e)

General Approach to Online Optimization Problems

Two Steps:• Generate in an online fashion a

fractional solution such that: Cost of online fractional solution is

close to cost of optimal fractional solution.

• Round the fractional solution online into an integral solution such that:

Cost of integral solution is close to cost of fractional solution.

First Part: Online Fractional Solution

Connectivity Problems:Optimal Cost – W*Cost of edges – [1, 2m2] (m = num. of

edges)Initially: Give each edge weight = 1/(2m3)

Total initial weight:• m edges• Maximal cost of edge – 2m2

Total initial cost at most 1

Algorithm – Online connectivity

• If maximum flow from S to T is at least 1: Do nothing

• Else: While the flow is less than 1:

1.Compute minimum cut C between S and T

2.For each edge e in the cut:w(e) w(e)[1+ 1/c(e)]

New demand D = (S, T):

The Algorithm - Analysis

Lemma: The total number of weight increments during the algorithm is O(W* logm)

Proof: Potential function:)(log2

*ee

Eee wwc

Algin edge of weight -

OPTin edge ofweight *

e

e

w

w

Analysis – cont.

• Initial value of the potential function is: -2W* log2(2m)

Initial weights of edges: we = 1/(2m3).

• The potential function never exceeds: 2W* The weight of each edge is at most 2.

• Each time weights are increased, the potential function increases by at least 1.

)(log2*

eeEe

e wwc

Analysis – cont.

Proof of third fact:

Cee

ee

Cee

eeCe

ee

eeCe

e

wc

wc

wwcc

wwc

11

1log

log1

1log

*2

*

2*

2*

)(log2*

eeEe

e wwc

• First inequality – (ce ≥ 1)• Second inequality – OPT is feasible.

The Algorithm – Competitive Ratio

Theorem: The algorithm is O(log m) competitive.

Proof:1. The initial value of the solution is at most 1.2. Each time the algorithm increases weights, the

cost it pays increases by:

c(e) w(e)/c(e) = w(e) ≤ 1

(The minimum cut is at most 1)

3. There are O(W* log m) weight increments in the algorithm.

Online Multicut - Algorithm

• If the shortest path from S to T is at least 1

Do nothing• Else:

While the distance is less than 1:1. Compute a shortest path P between S

and T2. For each edge e in the path P :

w(e) w(e)[1+ 1/c(e)]

New demand D = (S, T):

The Algorithm – Competitive Ratio

Theorem: The algorithm for generating a fractional multicut online is O(log m) competitive.

Proof:

Similar Analysis

Lower Bounds

Lemma: Any deterministic (and randomized) online algorithm for the fractional connectivity and fractional cuts problem has a competitive ratio of at least Ω(log m)

Remarks:1. Holds even with respect to the optimal

integral solution.

Rounding the Fractional SolutionThe rounding is problem specific.

Results:

1. Set cover, non-metric facility location and multicast – O(logn logm)- competitive algorithm.m – number of possible facilities.n – number of clients.

Remark: Lower bound for deterministic algorithm for online set cover – almost tight.

Rounding the Fractional Solution (cont.)

Results (cont.):

2. Online group Steiner Problem:1. Trees: O(logk log N logn)2. General Graphs: O(logk log N log2n)

n – number of vertices in the graphk – number of clientsN – maximal size of a group ( at most n)

Remark: General Graphs via HST’s

Rounding the Fractional Solution

Example: Online Set Cover Problem.

Offline case: Classic “randomized rounding”:

Choose each set S with probability O(w(S)logn):

• Elements are covered with high probability.

• Expected cost is fractional cost x O(logn).

Rounding the Fractional Solution

Online case: randomized rounding on the “increments” of the fractional increase.

In each weight augmentation:

w(S) w(S)[1+1/c(S)]Repeat O(logn) times: Choose Set S with

probability w(S)/c(S).

Surprisingly, this can be de-randomized online using a suitable potential function [AAABN, STOC ’03].

Rounding the Fractional SolutionExample: Multicast problem on trees• For each tree: choose 2logn’ r. v. uniformly

in [0,1]. (n’ – # terminals so far)• Threshold of a tree: minimum r. v.

Online Rounding: Take an edge if weight exceeds tree

threshold.(Weights on a path – monotone non-increasing)

Open: Can it be de-randomized? (even for facility location.)

Online Multicut Problem

Techniques:

1. Raecke’s hierarchical decomposition of a graph into a tree. (Harrelson, Hidrum, Rao).

2. Ratio of Minimum Cut / Maximum multi-commodity Flow in trees is at most 2.

3. Simple online primal-dual algorithm on trees.

Online Multicut Problem

Results: Deterministic online algorithm for the

multicut problem with competitive ratio:

• O(log3n loglogn) for general graphs.• O(log2n loglogn) for planar graphs.• O(log2n) for trees.

n – number of vertices

Thank you!