A Primal-Dual Approach to Online Optimization Problems

Online Optimization Problems

• input arrives “piece by piece” (“piece” is called request ) • upon arrival of a request - has to be served immediately • past decisions cannot be revoked how to evaluate the performance of an online algorithm?

– if, for each request sequence, cost(online) ≤ r x cost(optimal offline) – then online algorithm is r-competitive

Road Map

Introducing the framework:

• Ski rental • Online set cover • Virtual circuit routing

The general framework:

• {0,1} covering/packing linear programs • General covering/packing linear programs

Recent results:

• The ad-auctions problem • Weighted caching

The Ski Rental Problem

• Buying costs $B.

• Renting costs $1 per day.

Problem:

• Number of ski days is not known in advance.

Goal:

Minimize the total cost.

x

Ski Rental – Integer Program

 

1 - Buy 0 - Don't Buy

z i

 1 - Rent on day i  0 - Don't rent on day i Subject to:

min

Bx



i k

  1

z

i

For each day i:

x

 

i



i

1 {0,1}

Ski Rental – Relaxation

P: Primal Covering

min

Bx



i k

  1

z i

For each day i:

x

 

i

1  0

i

D: Dual Packing

max

i k

  1

y i

For each day i:

y i i k

  1

y i



B

 1 • • •

Online setting: Primal:

New constraints arrive one by one.

Requirement:

be satisfied. Upon arrival, constraints should

Monotonicity:

Variables can only be increased.

Ski Rental – Algorithm

P: Primal Covering

min

Bx



i k

  1

z i

For each day i:

x

 

i

1  0

i

D: Dual Packing

max

i k

  1

y i

For each day i:

i k

  1

y i y i

 1 

B

Initially x  0 Each new day (new constraint): if x<1:   z i  x  1-x x(1+ 1/B) + 1/(c*B)  y i  1 ‘c’ later.

Analysis of Online Algorithm

Proof of competitive factor: 1. Primal solution is feasible .

2. In each iteration, ΔP ≤ (1+ 1/c)ΔD . 3. Dual is feasible .

Conclusion:

Algorithm is (1+ 1/c)-competitive Initially x  0 Each new day (new constraint): if x<1:   z i  x  1-x x(1+ 1/B) + 1/(c*B)  y i  1 ‘c’ later.

Analysis of Online Algorithm

1. Primal solution is feasible.

If x ≥1 the solution is feasible.

Otherwise set: z i  1-x. • • 2. In each iteration, ΔP ≤ (1+ 1/c)ΔD: If x≥1, ΔP =ΔD=0

Algorithm:

Otherwise: Change in dual: 1 Change in primal: When new constraint arrives, if x<1: z i  1-x x  x(1+ 1/B) + 1/c*B y i  1 B Δx + z i = x+ 1/c+ 1-x = 1+1/c

Analysis of Online Algorithm

3.

Dual is feasible: Need to prove:

i k

  1

y i



B

We prove that after B days x≥1

Algorithm:

When new constraint arrives, if x<1: z i  1-x x  x(1+ 1/B) + 1/c*B y i  1 x is a sum of geometric sequence a 1 = 1/(cB), q = 1+1/B

x

 1

cB

  1  1  1

B

1

B B

 1   1 1  1

B c B

 1

c

   1

B



B

1

c e e

 1

e

1

• • •

Randomized Algorithm

X: 0 X 1 X 2 X 3 X 4 1 Choose

d

uniformly in [0,1] Buy on the day corresponding to the “bin”

d

falls in Rent up to that day • •

Analysis:

Probability of buying on the Probability of renting on the

i i

-th day is

x i

-th day is at most

z i

Key Idea for Primal-Dual

Primal: Min  i c i x i Dual: Max  t b t y t Step t , new constraint: a 1 x 1 + a 2 x 2 + … + a j x j ≥ b t x i  (1+ a i /c i ) x i ( mult.

update)  primal cost = New variable y t + b t y t in dual objective y t  y t + 1 ( additive update) =  Dual Cost

The Online Set-Cover Problem

• Elements: e 1 , e 2 , …, e n • Set system: s 1 , s 2 , … s m • Costs: c(s 1 ), c(s 2 ), … c(s m )

Online Setting:

• Elements arrive one by one. • Upon arrival elements need to be covered.

• Sets that are chosen cannot be “unchosen”.

Goal:

Minimize the cost of the chosen sets.

Set Cover – Linear Program

P: Primal Covering

min



E



D: Dual Packing

max 

S

  • • •

Online setting: Primal:

constraints arrive one by one.

Requirement: Monotonicity:

each constraint is satisfied. variables can only be increased.

P: Primal Covering

min



Set Cover – Algorithm

D: Dual Packing

max 

E



S

  Initially x(s)  When new element arrives, while  • y(e)  0 y(e)+1 • .

 

x s

    1/  

Analysis of Online Algorithm

Proof of competitive factor: 1. Primal solution is feasible .

2. In each iteration, ΔP ≤ 2ΔD . 3. Dual is (almost) feasible .

Conclusion:

We will see later.

Initially x(S)  0 When new element e arrives, while  • y(e)  y(e)+1 • .

 

x s

    1/  

Analysis of Online Algorithm

1. Primal solution is feasible.

We increase the primal variables until the constraint is feasible.

Initially x(S)  0 When new element e arrives, while  • y(e)  y(e)+1 • .

 

x s

    1/  

Analysis of Online Algorithm

2.

In each iteration,

ΔP ≤ 2ΔD.

• In each iteration: ΔD = 1    c(s) 

x s x

(s)   1/m   2  c(s)      1   Initially x(S)  0 When new element e arrives, while  • y(e)  y(e)+1 • .

 

x s

    1/  

• • • 3.

Analysis of Online Algorithm

Dual is (almost) feasible : We prove that:

s S

,  y(e)  c(s)O(log m) If y(e) increases, then

x(s) increases

(for e in S).

x(s) is a sum of a

geometric series

: a 1 = 1/[mc(s)], q = (1+ 1/c(s)) Initially x(S)  0 When new element e arrives, while  • y(e)  y(e)+1 • .

 

x s

    1/  

Analysis of Online Algorithm

 After c(s)O(log m) rounds:  1      

m



m

)   1    1  1

m

)  1  We

never increase a variable x(s)>1!

Initially x(S)  0 When new element e arrives, while  • y(e)  y(e)+1 • .

 

x s

    1/  

Conclusion

• •  The dual is feasible with cost 1/O(log m) of the primal. The algorithm produces a fractional set cover that is O(log m)-competitive.

Remark:

general.

No online algorithm can perform better in • • •

What about an integral solution?

Round fractional solution. (With O(log n) amplification.) Can be done deterministically online Competitive ratio is O(log m log n).

[AAABN03].

Online Virtual Circuit Routing

Network graph G=(V, E) capacity function u: E  Z +

Requests: r i = (s i , t i )

•

Problem:

request. Connect s i to t i by a path, or reject the • • Reserve one unit of bandwidth along the path.

No re-routing is allowed

.

• •

Load:

ratio between reserved edge bandwidth and edge capacity.

Goal:

Maximize the total throughput.

Routing – Linear Program

( , )

i

= Amount of bandwidth allocated for r i on path p

( )

i

- Available paths to serve request r i

max

 

r i

y r p

i

s.t: For each r i : 

y r p

i

 For each edge e:  

r i

y r p

i



P: Primal Covering

min   

r i



r p i



P r i

Routing – Linear Program

 x(e) 

z r i i



D: Dual Packing



r i

max   

r i y r p i y r p i

 

e

:  

r i y r p i

 • • •

Online setting: Dual:

new columns arrive one by one.

Requirement: Monotonicity:

each dual constraint is satisfied. variables can only be increased.

Routing – Algorithm

P: Primal Covering

min   

r i z r i



r p i



P r i

 x(e) 

i



D: Dual Packing



r i

max   

r i y r p i y r p i

 Initially x(e)  0 When new request arrives, if    .

z(r i ) 

e p

1 y(r i ,p)  1   ( ) 1  

e

:  

r i

1

P r i

    1 

y r p i



Analysis of Online Algorithm

Proof of competitive factor: 1. Primal solution is feasible .

2. In each iteration, ΔP ≤ 3ΔD . 3. Dual is (almost) feasible.

Conclusion:

We will see later.

Initially x(e)  0 When new request arrives, if   .

z(r i )   

p

 1 y(r i ,p)  1   ( ) 1  1

P r i

     1

Analysis of Online Algorithm

1. Primal solution is feasible.

i

 Otherwise: we update z(r i )  1 Initially x(e)  0 When new request arrives, if    .

z(r i ) 

e p

1 y(r i ,p)  1   ( ) 1 

P r i

1      1

Analysis of Online Algorithm

2.

In each iteration: ΔP ≤ 3ΔD.

 ΔP = ΔD=0 Otherwise: ΔD=1    ( )

i

    Initially x(e)  0 When new request arrives, if

P r i

   .

z(r i ) 

e p

1 y(r i ,p)  1   ( ) 1  1       1  1  

Analysis of Online Algorithm

3.

Dual is (almost) feasible . • We prove: For each e, after routing u(e)O(log n) on e, x(e)≥1 x(e) is a sum of a geometric sequence  x(e) 1 = 1/(nu(e)), q = 1+1/u(e) After u(e)O(log n) requests:   1    1  1   1    1    1  1    1  1  

n

 1  1

New Results via P-D Approach: Routing Previous results (routing/packing):

• •

[AAP93]

– Route O(log n) fraction of the optimal without violating capacity constraints.

Capacities must be

at least logarithmic

.

[AAFPW94]

– Route all the requests with load of at most O(log n) times the optimal load.

Observation [BN06]

– Both results can be described within the primal-dual approach.

New Results via P-D Approach: Routing

• We saw a simple algorithm which is:

3-competitive

and violates capacities by O(log n) factor.

•

Can be improved [Buchbinder, Naor, FOCS06] to: 1-competitive

and violates capacities by O(log n) factor.

Non Trivial.

Main ideas:

• Combination of ideas drawn from casting of previous routing algorithms within the primal-dual approach.

• Decomposition of the graph.

• Maintaining

several primal solutions

which are used to bound the dual solution, and for the routing decisions.

New Results via P-D Approach: Routing Applications [Buchbinder, N, FOCS 06]:

• Can be used as “black box” for many objective functions and in many routing models: –

Previous Settings

[AAP93,APPFW94].

–

Maximizing throughput.

–

Minimizing load.

–

Achieving better global fairness results (Coordinate competitiveness).

Road Map

Introducing the framework:

• Ski rental • Online set cover • Virtual circuit routing

The general framework:

• {0,1} covering/packing linear programs • General covering/packing linear programs

Recent results:

• The ad-auctions problem • Weighted caching

Online Primal-Dual Approach

• Can the

offline

problem be cast as a

covering/packing program

?

linear

• Can the online process be described as: –

New rows appearing in a covering LP?

–

New columns appearing in a packing LP?

Yes ??

• Upon arrival of a new request: – Update primal variables in a

multiplicative way

.

– Update dual variables in an

additive way

.

Online Primal Dual Approach

Next Prove:

1. Primal solution is

feasible

(or nearly feasible).

2. In each round,

ΔP ≤ c ΔD

.

3. Dual is

feasible

(or

nearly feasible

).

Got a

fractional

solution, but need an

integral

solution ??

• Randomized rounding techniques might work.

• Sometimes, even derandomization (e.g., method of conditional probabilities) can be applied online!

Online Primal-Dual Approach

Advantages: 1.

Generic

ideas and algorithms applicable to many online problems.

2.

3.

4.

5.

Linear Program

helps detecting the difficulties of the online problem.

General recipe

for the design and analysis of online algorithms.

No

potential function

appearing

“out of nowhere”

.

Competitiveness with respect to a

fractional optimal solution.

General Covering/Packing Results

What can you expect to get?

• For a {0,1} covering/packing matrix: –

Competitive ratio O(log D)

[BN05] (D – max number of non-zero entries in a constraint).

Remarks:

• Fractional solutions.

• Number of constraints/variables can be exponential.

• There can be a tradeoff between the competitive ratio and the factor by which constraints are violated.

General Covering/Packing Results

• For a

general covering/packing

matrix [BN05] :

Covering:

– Competitive ratio O(log n) (n – number of variables).

Packing:

– Competitive ratio O(log n + log [a(max)/a(min)]) a(max), a(min) – maximum/minimum non-zero entry

Remarks:

• Results are tight.

Special Cases

The max number of non-zero entries in a constraint is a constant?

• You can get a

constant ratio

.

The max number of non-zero entries in a constraint is 2?

• Calls for an

e/(e-1)-ratio.

Examples: • Ski rental, Online matching, Ad-Auctions.

Known Results via P-D Approach

• • • • •

Covering Online Problems (Minimization): O(log k)-algorithm for weighted caching Ski rental, Dynamic TCP Acknowledgement Parking Permit Problem Online Set Cover

[Meyerson 05] [AAABN03]

Online Graph Covering Problems

[BBN07] [AAABN04]: – Non-metric facility location – Generalized connectivity: pairs arrive online – Group Steiner: groups arrive online – Online multi-cut: (s,t)--pairs arrive online

Known Results via P-D Approach

• • • •

Packing Online Problems (maximization): Online Routing/Load Balancing Problems

AAPFW93, BN06 ].

[AAP93,

General Packing/routing e.g. Multicast trees.

Online Matching

[KVV91] – Nodes arrive one-by-one.

Ad-Auctions Problem

[MSVV05] – In a bit …

Road Map

Introducing the framework:

• Ski rental • Online set cover • Virtual circuit routing

The general framework:

• {0,1} covering/packing linear programs • General covering/packing linear programs

Recent results:

• The ad-auctions problem • Weighted caching

What are Ad-Auctions?

You type in a search engine: You get:

Vacation Eilat And … Advertisements Algorithmic Search results

How do search engines sell ads?

• • Each

advertiser

: – Sets a daily budget – Provides bids on interesting keywords

Search Engine

– Selects ads (on each keyword): – Advertiser pays bid if user clicks on ad.

Goal (of Search engine):

Maximize revenue

Mathematical Model

• Buyer i: – Has a daily budget B(i) • • Online Setting: – Items (keywords) arrive one-by-one.

– Each buyer gives a bid on each of the items (can be zero)

Algorithm:

– Assigns each item to some interested buyer.

Assumption:

Bids are small compared to the daily budget.

Ad-Auctions – Linear Program

I - Set of buyers.

J - Set of items.

B(i) – Budget of buyer i b(i,j) – bid of buyer i on item j s.t: For each item j: 

j-

th ad-auction is sold to buyer i.

max  

Each item is sold once.

exceed their budget

For each buyer i:  

Results

[ MSVV FOCS 05]: • (1-1/e)-competitive online algorithm.

• Bound is tight.

• Analysis uses

tradeoff revealing family of LP ’s

- not very intuitive.

Our Results [Buchbinder, Jain, N, 2007]:

• A different approach based on the primal-dual method:

very simple and intuitive

… and extensions.

• Techniques are applicable to many other problems.

The Paging/Caching Problem

Set of n pages, cache of size k

Request sequence of pages 1, 6, 4, 1, 4, 7, 6, 1, 3, … If requested page is in cache, no penalty .

Else, cache miss !

And load page into cache, (possibly) evicting some page. Goal: Minimize the number of cache misses.

Main Question: Which page to evacuate?

Previous Results: Paging

Paging (Deterministic) [Sleator Tarjan 85]: • Any det. algorithm >= k-competitive .

• LRU is k-competitive (also other algorithms) • LRU is k/(k-h+1)-competitive if optimal has cache of size h

• O(log(k/k-h+1))-competitive cache of size h

The Weighted Paging Problem

One small change:

• Each page i has a different load cost w(i).

• Models scenarios in which the cost of bringing pages is not uniform: Main memory, disk, internet …

web Goal

• Minimize the

total cost

of cache misses.

Weighted Paging (Previous Work)

Paging

Lower bound k

Weighted Paging

LRU k competitive k -competitive [Chrobak, Karloff, Payne, Vishwanathan 91] k/(k-h+1) if opt’s cache size h k/(k-h+1) [Young 94] O(log k) Marking Randomized O(log k/(k-h+1)) O(log k) [Irani 02] for two weight classes No o(k) algorithm known even for 3 weight classes.

The k-server Problem

• k servers lie in an n-point metric space .

• Requests arrive at metric points.

• To serve request: Need to move some server there.

Goal : Minimize total movement cost • Paging = k-server on a uniform metric .

(every page is a point, page in cache iff server on the point) • Weighted paging = k-server on a weighted star metric.

The k-server Problem

• k servers lie in an n-point metric space .

• Requests arrive at metric points.

• To serve request: Need to move some server there.

Goal : Minimize total movement cost (2k-1) Det. Work Function Alg [Koutsoupias, Papadimitriou 95] Randomized. No o(k) known (even for very simple spaces).

Best lower bound  (log k) (widely believed conjecture)

Our Results

Weighted Paging (Randomized): (Bansal, Buchbinder, N., FOCS 2007) • O(log k)-competitive algorithm for weighted paging.

• O(log (k/k-h+1))-competitive if opt ’s cache size h

Much simpler than previous approaches.

Metrical Task System (Randomized): • O(log N)-competitive algorithm on a weighted star metric.

• Closely related to k-server problem (details in paper …)

Further Research

Generalized Caching

• Pages have both sizes and fetching costs .

• Motivation: Web-Caching

Special models:

• Bit model: Fetching cost proportional to size (minimize traffic) • Fault model: Fetching cost is uniform (minimize number of times a user has to wait for a page)

Results

(Bansal, Buchbinder, N., 2008): • O(log k) competitive algorithms for Bit and Fault models.

• O(log 2 k) competitive algorithm for the general model .

• Requires new interesting ideas and interesting analysis in the rounding phase.

Further Research

• More applications.

• Extending the general framework beyond packing/covering.

• The

k

-server problem?

Thank you