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

Analysis of Online Algorithm

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

Analysis of Online Algorithm

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

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) 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

Analysis of Online Algorithm

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

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

Analysis of Online Algorithm

2. In each iteration, ΔP ≤ 2ΔD.

In each iteration:

• ΔD = 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

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

Analysis of Online Algorithm

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))

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

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

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) 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

Analysis of Online Algorithm

1. Primal solution is feasible.

If the solution is feasible.

Otherwise: we update z(ri) 1e p

( ), x(e) 1:ip P 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

Analysis of Online Algorithm

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

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

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:( ) (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).

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

Online Primal-Dual Approach

• Can the offline problem be cast as a linear covering/packing program?

• 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. Linear Program helps detecting the difficulties of the online problem.

3. General recipe for the design and analysis of online algorithms.

4. No potential function appearing “out of nowhere”.

5. 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 [BBN07]

• Ski rental, Dynamic TCP Acknowledgement

• Parking Permit Problem [Meyerson 05]

• Online Set Cover [AAABN03]• Online Graph Covering Problems [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 [AAP93, AAPFW93, BN06].

• General Packing/routing e.g. Multicast trees.

• Online Matching [KVV91] – Nodes arrive one-by-one.

• Ad-Auctions Problem [MSVV05] – In a bit …

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

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