a primal-dual approach to online optimization problems

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 MapIntroducing 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.

Ski Rental – Integer Program

Subject to:

For each day i:

1 - Rent on day i

0 - Don't rent on day i iz

1 - Buy

0 - Don't Buyx

1

mink

ii

Bx z

1ix z , {0,1}ix z

D: Dual Packing

For each day i:

Ski Rental – RelaxationP: Primal Covering

For each day i: 1

mink

ii

Bx z

1iy

, 0ix z

1

maxk

ii

y

1ix z

1

k

ii

y B

Online setting:

• Primal: New constraints arrive one by one.

• Requirement: Upon arrival, constraints should be satisfied.

• Monotonicity: Variables can only be increased.

D: Dual Packing

For each day i:

Ski Rental – AlgorithmP: Primal Covering

For each day i: 1

mink

ii

Bx z

1iy

, 0ix z

1

maxk

ii

y

1ix z

1

k

ii

y B

Initially x 0Each new day (new constraint): if x<1:

zi 1-x x x(1+ 1/B) + 1/(c*B) - ‘c’ later. yi 1

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 0Each new day (new constraint): if x<1:

zi 1-x x x(1+ 1/B) + 1/(c*B) - ‘c’ later. yi 1


1. Primal solution is feasible.If x ≥1 the solution is feasible.

Otherwise set: zi 1-x.

2. In each iteration, ΔP ≤ (1+ 1/c)ΔD: If x≥1, ΔP =ΔD=0Otherwise: • Change in dual: 1• Change in primal:

BΔx + zi = x+ 1/c+ 1-x = 1+1/c

Algorithm:When new constraint

arrives, if x<1:

zi1-xx x(1+ 1/B) + 1/c*B

yi1


3. Dual is feasible:

Need to prove:

We prove that after B days x≥1

x is a sum of geometric sequence

a1 = 1/(cB), q = 1+1/B

1

k

ii

y B

1 11 1 1 1

11

1 1

B B

B BxcB c

B

11 1 1

B

c eB

1

11

e

c e

Algorithm:When new constraint

arrives, if x<1:

zi1-xx x(1+ 1/B) + 1/c*B

yi1

Randomized Algorithm

• 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 i-th day is xi

• Probability of renting on the i-th day is at most zi

X1 X2 X3 X4

0 1X:

Key Idea for Primal-Dual

Primal: Min i ci xi Dual: Max t bt yt

Step t, new constraint: New variable yt

a1x1 + a2x2 + … + ajxj ≥ bt + bt yt in dual objective

xi (1+ ai/ci) xi (mult. update) yt yt + 1 (additive update)

primal cost =

= Dual Cost

The Online Set-Cover Problem

• Elements: e1, e2, …, en

• Set system: s1, s2, … sm

• Costs: c(s1), c(s2), … c(sm)

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.

D: Dual Packing

Set Cover – Linear ProgramP: Primal Covering

Online setting:

• Primal: constraints arrive one by one.

• Requirement: each constraint is satisfied.

• Monotonicity: variables can only be increased.

max ( )e E

y e

( ) ( )e s

s S y e c s

min ( ) ( )s S

c s x s ( ) 1s e s

e E x s

D: Dual Packing

Set Cover – AlgorithmP: Primal Covering

max ( )e E

y e

( ) ( )e s

s S y e c s

min ( ) ( )s S

c s x s ( ) 1s e s

e E x s

Initially x(s) 0When new element arrives, while

• y(e) y(e)+1• .

( ) 1:s e s

x s

s e s ( ) ( ) 1 1/ ( ) 1/ ( )x s x s c s m c s




2. In each iteration, ΔP ≤ 2ΔD.

3. Dual is (almost) feasible.

Conclusion: We will see later.

Initially x(S) 0When new element e arrives, while

• y(e) y(e)+1

• .

( ) 1:s e s

x s

s e s ( ) ( ) 1 1/ ( ) 1/ ( )x s x s c s m c s


1. Primal solution is feasible.We increase the primal variables until the constraint is feasible.


• y(e) y(e)+1

• .

( ) 1:s e s

x s

s e s ( ) ( ) 1 1/ ( ) 1/ ( )x s x s c s m c s



In each iteration:

• ΔD = 1


• y(e) y(e)+1

• .

( ) 1:s e s

x s

s e s ( ) ( ) 1 1/ ( ) 1/ ( )x s x s c s m c s

s|e s s|e s

(s) 1/m 2x

s|e s s|e s

( ) 1 P = c(s) ( ) c(s)

( ) ( )

x sx s

c s m c s


3. Dual is (almost) feasible:

• We prove that:

• If y(e) increases, then x(s) increases (for e in S).

• x(s) is a sum of a geometric series:

a1 = 1/[mc(s)], q = (1+ 1/c(s))


• y(e) y(e)+1

• .

( ) 1:s e s

x s

s e s ( ) ( ) 1 1/ ( ) 1/ ( )x s x s c s m c s

e S

, y(e) c(s)O(log m)s S

Analysis of Online Algorithm After c(s)O(log m) rounds:

We never increase a variable x(s)>1!Initially x(S) 0When new element e arrives, while

• y(e) y(e)+1

• .

( ) 1:s e s

x s

s e s ( ) ( ) 1 1/ ( ) 1/ ( )x s x s c s m c s

( ) (log )1 1/ ( ) 11

( )( ) 1 1/ ( ) 1

c s O mc s

x sm c s c s

( ) (log )1 1/ ( ) 1

1

c s O mc s

m

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: No online algorithm can perform better in general.

What about an integral solution?

• Round fractional solution. (With O(log n) amplification.)

• Can be done deterministically online [AAABN03].

• Competitive ratio is O(log m log n).

Online Virtual Circuit Routing

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

Requests: ri = (si, ti)

• Problem: Connect si to ti by a path, or reject the request.

• 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

s.t:

For each ri:

For each edge e:

( , )iy r p

( )

max ( , )i i

ir p P r

y r p

= Amount of bandwidth allocated for ri on path p

( )iP r - Available paths to serve request ri

( )

( , ) 1i

ip P r

y r p

( )

( , ) ( )i i

ir p P r e p

y r p u e

D: Dual Packing

Routing – Linear ProgramP: Primal Covering

min ( ) ( ) ( )i

ie E r

u e x e z r

, ( ) :i ir p P r

Online setting:• Dual: new columns arrive one by one.• Requirement: each dual constraint is satisfied. • Monotonicity: variables can only be increased.

( )

max ( , )i i

ir p P r

y r p

( )

( , ) 1i

i ip P r

r y r p

( )

: ( , ) ( )i i

ir p P r e p

e y r p u e

e p

x(e) ( ) 1iz r

D: Dual Packing

Routing – AlgorithmP: Primal Covering

min ( ) ( ) ( )i

ie E r

u e x e z r

, ( ) :i ir p P r

( )

max ( , )i i

ir p P r

y r p

( )

( , ) 1i

i ip P r

r y r p

( )

: ( , ) ( )i i

ir p P r e p

e y r p u e

e p

x(e) ( ) 1iz r

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

z(ri) 1 . y(ri,p) 1

e p

( ), x(e) 1:ip P r

1 1

: ( ) ( ) 1( ) ( )

e p x e x eu e n u e






Conclusion: We will see later.


z(ri) 1 . y(ri,p) 1

e p

( ), x(e) 1:ip P r

1 1

: ( ) ( ) 1( ) ( )




If the solution is feasible.

Otherwise: we update z(ri) 1e p

( ), x(e) 1:ip P r


z(ri) 1 . y(ri,p) 1

e p

( ), x(e) 1:ip P r

1 1

: ( ) ( ) 1( ) ( )



2. In each iteration: ΔP ≤ 3ΔD.

If ΔP = ΔD=0

Otherwise:

ΔD=1

e p

( ) : x(e) 1ip P r

( ) ( ) ( )ie p

P u e x e z r

( ) 1( ) 1 3

( ) ( )e p

x eu e

u e n u e


z(ri) 1 . y(ri,p) 1

e p

( ), x(e) 1:ip P r

1 1

: ( ) ( ) 1( ) ( )




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:( ) (log ) ( ) (log )

1 11 1 1 1

( ) ( )1 ( ) 1

( ) 11 1

( )

u e O n u e O n

u e u ex e

n u e n

u e

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: RoutingWe 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).





Recent results:


What are Ad-Auctions?

You type in a search engine:

You get:

AlgorithmicSearch results

And …Advertisements

Vacation Eilat

How do search engines sell ads?

• Each advertiser: – Sets a daily budget – Provides bids on interesting keywords

• Search Engine (on each keyword):– Selects ads– 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

s.t:

For each item j:

For each buyer i:

( , )y i j

max ( , ) ( , )i I j J

b i j y i j

I - Set of buyers.J - Set of items.

( , ) 1i I

y i j

= 1 j-th ad-auction is sold to buyer i.

( , ) ( , ) ( )j J

b i j y i j B i

B(i) – Budget of buyer ib(i,j) – bid of buyer i on item j

Each item is sold once.Buyers do not

exceed their budget

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<n.

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: PagingPaging (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<k

Paging (Randomized):

• Rand. Marking O(log k) [Fiat, Karp, Luby, McGeoch, Sleator, Young 91].

• Lower bound Hk [Fiat et al. 91], tight results known.

• O(log(k/k-h+1))-competitive algorithm if optimal hascache of size h<k [Young 91]

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 …

Goal • Minimize the total cost of cache misses.

web

Weighted Paging (Previous Work)

Lower bound k

LRU k competitive

k/(k-h+1) if opt’s cache size h

k-competitive [Chrobak, Karloff, Payne, Vishwanathan 91]

k/(k-h+1) [Young 94]

O(log k) Randomized Marking

O(log k/(k-h+1))

O(log k) for two weight classes [Irani 02]

No o(k) algorithm known even for 3 weight classes.

Paging Weighted Paging

De

term

inis

tic

Ra

nd

om

ize

d

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<k.

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 ResearchGeneralized 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(log2k)-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

a primal-dual approach to online optimization problems

Documents

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