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From the marriage problem to sailor assignment and Google AdWords problems

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From the marriage problem to sailor assignment and Google AdWords problems

MATCHING PROBLEMS OLD AND NEW:

G. Hernandez

UofM / UNAL Colombia

- Based on:
- “The Mathematics Of 1950’s Dating: Who wins The Battle of The Sexes?”` from Great Theoretical Ideas in Computer Science http://www.cs.cmu.edu/~15251/index.html
- “New Market Models and Algorithms'‘ talk form V. Vazirani http://www.cc.gatech.edu/~vazirani/

- Matching , Assignment and Correspondence Problems
- The Stable Marriage Problem
Gale-Shapley Algorithm

- Linear Assignment Problem
Kuhn-Munkres Algorithm

- The Sailor Assignment Problem
KM extension

EMOAs, HMOAs

- The Google AdWords Problem

There are n (agents, boys, sailors, bets, …)

and n (tasks, girls, jobs, adwords, …)

We may have restrictions on the agents can

perform the particular tasks

We may have preference lists or a cost

of assigning an agent to a task, or both

Question: How do we pair, match, assign, make them correspond off in an optimal way ?

The Traditional Marriage Problem

1

2

3

Taken from http://www.cs.cmu.edu/~15251/index.html

Dating Scenario

There are n boys and n girls

Each girl has her own ranked preference list of all the boys

Each boy has his own ranked preference list of the girls

The lists have no ties

Question: How do we pair them off?

3,2,5,1,4

3,5,2,1,4

1

1

5,2,1,4,3

1,2,5,3,4

2

2

4,3,5,1,2

4,3,2,1,5

3

3

1,2,3,4,5

1,3,4,2,5

4

4

2,3,4,1,5

1,2,4,5,3

5

5

More Than One Notion of What Constitutes A “Good” Pairing

Maximizing total satisfaction

Hong Kong and to an extent the USA

Maximizing the minimum satisfaction

Western Europe

Minimizing maximum difference in mate ranks

Sweden

Maximizing people who get their first choice

Barbie and Ken Land

Rogue Couples - Instability

Suppose we pair off all the boys and girls

Now suppose that some boy and some girl prefer each other to the people to whom they are paired

They will be called a rogue couple

Why be with them when we can be with each other?

What use is fairness, if it is not stable?

- Any list of criteria for a good pairing for “marriage” must include stability.
- If the pairing can be challenged in an open marketit is doomed if it contains a rogue couple.

Stable Pairings

A pairing of boys and girls is called stable if it contains no rogue couples

Stable Pairings

A pairing of boys and girls is called stable if it contains no rogue couples

3,2,1

3,2,1

1

1

2,1,3

1,2,3

2

2

3,1,2

3,2,1

3

3

Worshipping Males

The Traditional Marriage Algorithm

Female

String

The Traditional Marriage Algorithm

Gale-Shapley

For each day that some boy gets a “No” do:

Morning

- Each girl stands on her balcony

- Each boy proposes to the best girl whom he has not yet crossed off

Afternoon (for girls with at least one suitor)

- To today’s best: “Maybe, return tomorrow”

- To any others: “No, I will never marry you”

Evening

- Any rejected boy crosses the girl off his list

If no boys get a “No”, each girl marries boy to whom she just said “maybe”

The Traditional Marriage Algorithm

3,1,2

3,2,1

1

1

2,1,3

1,2,3

2

2

3,1,2

3,2,1

3

3

The Traditional Marriage Algorithm

3,1,2

3,2,1

1

1

2,1,3

1,2,3

2

2

3,1,2

3,2,1

3

3

The Traditional Marriage Algorithm

X

3,1,2

3,2,1

1

1

2,1,3

1,2,3

2

2

3,1,2

3,2,1

3

3

The Traditional Marriage Algorithm

X

3,1,2

3,2,1

1

1

2,1,3

1,2,3

2

2

3,1,2

3,2,1

3

3

The Traditional Marriage Algorithm

X

X

3,1,2

3,2,1

1

1

2,1,3

1,2,3

2

2

3,1,2

3,2,1

3

3

The Traditional Marriage Algorithm

X

X

3,1,2

3,2,1

1

1

2,1,3

1,2,3

2

2

3,1,2

3,2,1

3

3

The Traditional Marriage Algorithm

X

X

3,1,2

3,2,1

1

1

2,1,3

1,2,3

2

2

3,1,2

3,2,1

3

3

The Traditional Marriage Algorithm

3,1,2

3,2,1

1

1

2,1,3

1,2,3

2

2

3,1,2

3,2,1

3

3

The TMA always terminates in at most n2 days.

A “master list” of all n of the boys lists starts with a total of n x n = n2girls on it

Each day that at least one boy gets a “No”, so at least one girl gets crossed off the master list

Therefore, the number of days is bounded by the original size of the master list

TMA always produces a male-optimal, female-pessimal pairing.

A boy’s optimal girl is the highest ranked girl for whom there is some stable pairing in which the boy gets her

TMA Related problems!

- Weighted matching
- Stable roommates
- hospitals/residents problem
- hospitals/residents problem with couples

Current real applications of TMA!

National Resident Matching Program

http://www.nrmp.org

The National Resident Matching Program (NRMP) is a private, not-for-profit corporation established in 1952 to provide a uniform date of appointment to positions in graduate medical education (GME) in the United States.

r1,1r1,2r1,3… r1,n

r2,1r2,2r2,3… r2,n

:

rn,1rn,2 rn,3 … rn,n

Tasks

Agents

r0,0

1

1

.

.

.

.

.

.

.

n! assignments

2

2

3

3

.

.

.

.

rn,n

n

n

Complete Bipartite Graph

LAP Kuhn-Munkres Algorithm

The Kuhn- Munkers (Hungarian) algorithm is used to find an optimum (maximum or minimum weight) matching in a complete bipartite graph in time O(n3 )

Step 1: For each row of the matrix, find the smallest element and subtract it from every element in its row.

LAP Kuhn-Munkres Algorithm

Step 2: Find a zero (Z) in the resulting matrix. If there is no starred zero in its row or column, star Z. Repeat for each element in the matrix.

Step 3: Cover each column containing a starred zero. If n columns are covered, the starred zeros describe a complete set of unique assignments. In this case STOP.

LAP Kuhn-Munkres Algorithm

Step 2: Find a zero (Z) in the resulting matrix. If there is no starred zero in its row or column, star Z. Repeat for each element in the matrix.

Step 3: Cover each column containing a starred zero. If n columns are covered, the starred zeros describe a complete set of unique assignments. In this case STOP.

LAP Kuhn-Munkres Algorithm

Step 4: Find a uncovered zero and prime it. If there is no starred zero in the row containing this primed zero, Go to Step 5. Otherwise, cover this row and uncover the column containing the starred zero. Continue in this manner until there are no uncovered zeros left. Save the smallest uncovered value and Go to Step 6.

Step 5: Construct a series of alternating primed and starred zeros as follows. Let Z0 represent the uncovered primed zero found in Step 4. Let Z1 denote the starred zero in the column of Z0 (if any). Let Z2 denote the primed zero in the row of Z1 (there will always be one). Continue until the series terminates at a primed zero that has no starred zero in its column. Unstar each starred zero of the series, star each primed zero of the series, erase all primes and uncover every line in the matrix. Return to Step 3.

Step 6: Add the value found in Step 4 to every element of each covered row, and subtract it from every element of each uncovered column. Return to Step 4 without altering any stars, primes, or covered lines.

LAP Kuhn-Munkres Algorithm

Step: 4

Step: 6

Step: 4

Step: 5

Step: 3

Final Solution

Sailors

120,000/year

1,000-10,000 to be assigned every two weeks

Commanders

Jobs database

XMLDB

Four objectives

TS

PCS

SR

CR

Detailers

200

MK algorithm applied to SAP

Problems

- SAP is multi-objective whereas LAP is single-objective.
- Solution: use of weight vectors
- In case of SAP, we do not have a complete bipartite graph (many infinite values for big instances with).
- Solution: sparse matrix represent.
- Sometimes we do not have a complete matching (conflicts), in those case MK does not work.
- Solution: The use of dummy jobs.

Sailors

Jobs

1

1

2

2

3

3

4

4

5

5

6

7

dummy

Job ID

Sailor-Job Information

r0,0r0,1 … rd0 …

r1,0r1,1 … rd1 …

: :

rn,0rn, 1 … … rdn

r0,0r0,1 …

r1,0r1,1 …

:

rn,0rn, 1 …

…

k

2

1

….

r1,k

r1,2

r1,1

….

….

1

Sailor Array

(List)

2

FTS1,1

3

FPCS1,1

Reduced rating

.

FSR1,1

.

Weight vectors

FCR1,1

.

n

Dummy jobs

Sparse Matrix representation

After adding dummy jobs in the matrix

RESULTS

KM(287)

NSGAII

NGSA(300)-KM (5)

Web

QUERY

- Yahoo!
- MSN

Paid for each “impression”

links!

Pais when someone clicks on

Advertisers

- Monika Henzinger, 2004: Find on-line algorithm to maximize Google’s revenue.
- Queries are coming on-line. Instantaneously decide which bidder gets it.

The Adwords Problem

N advertisers;

- Daily Budgets B1, B2, …, BN
- Each advertiser provides bids for keywords he is interested in.

Search Engine

N advertisers;

- Daily Budgets B1, B2, …, BN
- Each advertiser provides bids for keywords he is interested in.

Search Engine

queries

(online)

N advertisers;

- Daily Budgets B1, B2, …, BN
- Each advertiser provides bids for keywords he is interested in.

Search Engine

Select one Ad

Advertiser

pays his bid

queries

(online)

N advertisers;

- Daily Budgets B1, B2, …, BN
- Each advertiser provides bids for keywords he is interested in.

Search Engine

Select one Ad

Advertiser

pays his bid

queries

(online)

Maximize total revenue

Online competitive analysis - compare with best offline allocation

N advertisers;

- Daily Budgets B1, B2, …, BN
- Each advertiser provides bids for keywords he is interested in.

Search Engine

Select one Ad

Advertiser

pays his bid

queries

(online)

Maximize total revenue

Example – Assign to highest bidder: only ½ the offline revenue

Algorithm Greedy

Bidder 1 Bidder 2

Bidder1

Bidder 2

Book

Queries: 100 Books then 100 CDs

CD

B1 = B2 = $100

LOST

Revenue

100$

Optimal Allocation

Bidder 1 Bidder 2

Bidder1

Bidder 2

Book

Queries: 100 Books then 100 CDs

CD

B1 = B2 = $100

Revenue

199$

- Each daily budget is $1, and
each bid is $0/1.

queries

(girls)

advertisers

(boys)

queries

(girls)

advertisers

(boys)

queries

(girls)

advertisers

(boys)

queries

(girls)

advertisers

(boys)

queries

(girls)

advertisers

(boys)

queries

(girls)

advertisers

(boys)

queries

(girls)

advertisers

(boys)

- Karp, Vazirani & Vazirani, 1990:
Online optimal (1-1/e =0.63)n matching .

- 2n nodes bipartite graph
- Perfect matching (n pairs).
- Any deterministic online algorithm has expected value <=
- RANDOM (adversary) 1/2n

Online bipartite matching

1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0

- .
- .
- .
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

- .
- .
- .
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

queries

(girls)

advertisers

(boys)

n/2

:

:

n/2

:

:

queries

(girls)

advertisers

(boys)

n/2

:

:

n/2

:

:

advertisers

(boys)

queries

(girls)

- Any deterministic online algorithm has expected value on this instance <= (1-1/e)n

n/2

:

:

:

:

- Karp, Vazirani & Vazirani, 1990:
- RANKING:
- Rank by random permutation the boys vertices at the beging
- As each girl arrives, match her to the eligible boy, if any, of highest rank.
Expected number of pairs n (1-1/e) + o(n)

Optimal.

- RANKING:

- Kalyanasundaram & Pruhs, 2000:
On-line deterministic algorithm:

1-1/e factor algorithm for b-matching:

Daily budgets $b, bids $0/1, b>>1

- Awards the query to the interested
advertiser who has the highest unspent budget.

- Awards the query to the interested

- Mehta, Saberi, Vazirani & Vazirani, 2005:
1-1/e algorithm, assuming

budgets>>bids.

bids = 0, 1 or 2.

- Natural extension of KP00 that assigns the query to the highest bidder, in case of ties: the largest remaining budget. strictly smaller and Bounded Away from 1-1/e.

- Mehta, Saberi, Vazirani & Vazirani, 2007:
1-1/e algorithm, assuming budgets>>bids.

- A bidder pays only if the user clicks on his ad.
- Advertisers have different daily budgets.
- Instead of charging a bidder his actual bid, the search engine company charges him the next highest bid.
- Multiple ads can appear with the results of a query.
- Advertisers enter at different times.
Optimal!

- Idea: Use both bid and
fraction of left-over budget

- Idea: Use both bid and
fraction of left-over budget

- Correct tradeoff given by
tradeoff-revealing family of LP’s

The optimal tradeoff function is:

ψ(x) = 1 – e -x

- Allocate the next query to the bidder i maximizing the product of his bid and ψ(T(i)), where T(i) is the fraction of the bidder´s budget which has been spent so far, i.e., T(i) = mi / bi , where bi is the total budget of bidder i, mi is the amount of money spent by bidder i when the query arrives.
The algorithm assumes that the daily budget of advertisers is large compared to their bids.

- The randomized algorithm CHOSES a random permutation of ADVERTISERS in advance, and rank the ads in order of
Where d is the bid and n is the total number of advertisers and rank is the advertiser’s rank in the permutation

d*ψ(rank / n)

Google, then, will maximize its profits if it allocates queries to the advertiser with the highest value of d*ψ(f), where d is the bid and f is rank of the fraction of the budget that has not yet been spent.

- The current algorithm assumes that Google has no advance information about each day´s sequence of queries when, in fact, it does.
- Theoretic Angle: A ranking algorithm should minimize manipulation of the market by encouraging advertisers to reveal their true valuations for a given keyword and their true daily budgets.

- An inherently algorithmic theory of
market equilibrium

- New models that capture new markets

- Beginnings of such a theory, within
Algorithmic Game Theory

- Started with combinatorial algorithms
for traditional market models

- New market models emerging

- Strongly poly algs for approximating
- nonlinear convex programs
- equilibria

- Insights into congestion control protocols?

- Borgs et al, gave some evidence that it is impossible to design a truthful mechanism in the presence of budget constraints. [BCI+05]

- D. Gale and L. S. Shapley: College Admissions and the Stability of Marriage, American Mathematical Monthly 69, 9-14, 1962.
- R.M. Karp, U.V. Vazirani, and V.V. Vazirani. An optimal algorithm for online bipartite matching. In proceedings of the 22nd Annual ACM Symposium on Theory of Computing, 1990.
- Mehta, A. Saberi, A. Vazirani, U. Vazirani, V. AdWords and Generalized On-line Matching. Journal of the ACM 2007.
- C. Borgs, J. Chayes, N. Inmmorlica, M. Mahdian, and A. Saberi. Multi-unit auctions with budget-constrained bidders. In ACM conference on electronic commerce. 2005.
- B. Kalyanasundaram and K.R. Pruhs. An optimal deterministic algorithm for online b -matching. Theoretical Computer Science, 233(1{2):319{-25, 2000.
- Computer Scientists Optimize Innovative Ad Auction, SIAM News, Volume 38, Number 3, April 2005.