Graduate AI Lecture 8: Integer Programming Applications - - PowerPoint PPT Presentation

Graduate AI

Lecture 8: Integer Programming Applications

Instructors: Nihar B. Shah (this time)

• J. Zico Kolter

15780 Spring 2019: Lecture 2

APPLICATION: REVIEWER ASSIGNMENT IN PEER REVIEW

• d papers, n reviewers
• Each reviewer can review at most μ papers
• Each paper must be reviewed by at least λ

reviewers

• Given: Similarity scores between every paper-

reviewer pair:

• For every pair (Paper !, Reviewer "), similarity score #$% ∈ [(, *]
• Higher similarity score ⇒ Better envisaged quality of review

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15780 Spring 2019: Lecture 2

MAXIMIZING SUM SIMILARITIES

• Assignment in many conferences aims to
• ptimize the “sum similarity”
• Maximize sum of similarities of all assigned

(reviewer, paper) pairs

Nihar B. Shah, CMU 3

Note: There are other constraints such as conflicts of interest, but these are easy to incorporate and we will ignore them here.

15780 Spring 2019: Lecture 2

MAXIMIZING SUM SIMILARITIES

maximize '

( ∈ *+,-./

'

0 ∈1-23-4-./

5(0 6 paper i assigned to reviewer j

subject to every paper gets at least λ reviewers every reviewer gets at most μ papers

Nihar B. Shah, CMU 4

15780 Spring 2019: Lecture 2

INTEGER PROGRAM

maximize

' ∈)* × ,

• . ∈[0]
• 2 ∈[3]

4.2 5.2

subject to ∑ .∈[3] 5.2 ≤ 8 ∀ : ∑. ∈[3] 5.2 ≥ < ∀ = 5.2 ∈ 0,1 ∀ =, :

Nihar B. Shah, CMU 5

15780 Spring 2019: Lecture 2

TOTALLY UNIMODULAR MATRICES

Nihar B. Shah, CMU 6

Definition: A matrix is called a Totally Unimodular Matrix (TUM) if every square submatrix has a determinant -1, 0, or 1.

15780 Spring 2019: Lecture 2

TOTALLY UNIMODULAR MATRICES

Nihar B. Shah, CMU 7

Theorem: Consider a linear program with constraint Ax ≤ b. If A is TUM and b has integer entries then all vertices of the feasible set are integers.

Intuition: Recall Cramer’s rule. The system of equations Ax = b for square, non-singular A has solution !" = $%&(( ) )

$%&(() where + " is the

matrix obtained by replacing the ith column of A with b.

Definition: A matrix is called a Totally Unimodular Matrix (TUM) if every square submatrix has a determinant -1, 0, or 1.

15780 Spring 2019: Lecture 2

HOMEWORK

8

maximize

' ∈)* × ,

• . ∈[0]
• 2 ∈[3]

4.2 5.2

subject to ∑ .∈[3] 5.2 ≤ 8 ∀ : ∑. ∈[3] 5.2 ≥ < ∀ = 5.2 ∈ 0,1 ∀ =, :

Is the constraint in the following problem TUM?

15780 Spring 2019: Lecture 2

BACK TO REVIEWER ASSIGNMENT

A concern with current popular approach: (lack of) Fairness

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Paper 1 Paper 2 Reviewer 1 1 0.01 Reviewer 2 0.7 0.5 Reviewer 3 0.1 0.01 Reviewer 4 0.1 0.01 Suppose ! = 1, % = 2 Similarities:

• Assigns Reviewers 1, 2 to Paper 1; Reviewers 3,4 to Paper 2
• Quite bad for paper 2!
• Better solution: Reviewers 2, 4 to Paper 2

15780 Spring 2019: Lecture 2

A MORE FAIR APPROACH

maximize '

( ∈ *+,-./

'

0 ∈1-23-4-./

5(0 6 paper i assigned to reviewer j

Nihar B. Shah, CMU 10

maximize min

( ∈ *+,-./

'

0 ∈1-23-4-./

5(0 6 paper i assigned to reviewer j

[Stelmakh et al. 2019]

15780 Spring 2019: Lecture 2

A MORE FAIR APPROACH

Nihar B. Shah, CMU 11

maximize

' ∈)* × , min . ∈[0] 2 3 ∈[4]

5.3 6.3

such that ∑ .∈[4] 6.3 ≤ 9 ∀ ; ∑. ∈[4] 6.3 ≥ = ∀ > 6.3 ∈ 0,1 ∀ >, ;

This requires maximization of a minimum. Can we still use integer programming?

15780 Spring 2019: Lecture 2

A MORE FAIR APPROACH

Nihar B. Shah, CMU 12

maximize

' ∈)* × ,

• such that

∑/ ∈[1] 34/ 54/ ≥ - ∀ 8 ∑ 4∈[1] 54/ ≤ : ∀ ; ∑4 ∈[1] 54/ ≥ < ∀ 8 54/ ∈ 0,1 ∀ 8, ;

15780 Spring 2019: Lecture 2

APPLICATION: SUDOKU

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4 6 7 4 6 5 8 1 3 7 2 7 8 1 3 5 9 4 8 9 2 9 5 5 2 1 3

15780 Spring 2019: Lecture 2

SUDOKU

• For each !, #, $ ∈ [9], binary variable )*

+, s.t. )* +, = 1

iff we put $ in entry (!, #)

• For t = 1, … , 27, 56 is a row, column, or 3×3 square

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find )>

>>, … , )? ??

s.t. ∀C ∈ 27 , ∀$ ∈ [9], ∀!, #, $ ∈ [9], )*

+, ∈ {0,1}

∑ +,, ∈HI )*

+, = 1

∀!, # ∈ 9 , ∑*∈[?] )*

+, = 1

15780 Spring 2019: Lecture 2

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Note: If you have a hard time expressing something as an IP, try using binary variables.

15780 Spring 2019: Lecture 2

APPLICATION: ENVY-FREENESS

• Players ! = {1, … , '} and items ) = {1, … , *}
• Player + has value ,-. for item /
• Partition items to bundles 01, … , 02
• 01, … , 02 is envy-free iff ∀+, +4, ∑.∈78 ,-. ≥ ∑.∈78: ,-.

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$30 $50 $2 $5 $5 $3 $5 $2 $10 $5 $20 $20 $3 $40

1 2 1 2

15780 Spring 2019: Lecture 2

ENVY-FREENESS

• Variables: !"# ∈ 0,1 , !"# = 1 iff ) ∈ *"
• ENVY-FREE as an IP:

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find !//, … , !12 s.t. ∀7 ∈ 8, ∀79 ∈ 8, ∀) ∈ :, ∀7 ∈ 8, ) ∈ :, !"# ∈ {0,1} ∑#∈> ?"#!"# ≥ ∑#∈> ?"#!"A# ∑"∈B !"# = 1

15780 Spring 2019: Lecture 2

PHASE TRANSITION

• Imagine the !"# are drawn independently and

uniformly at random from [0,1]

• Question 1: If ) = +/2, what is the

probability that an envy-free allocation exists?

1. 2.

2/+

3.

1/2

4.

1

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15780 Spring 2019: Lecture 2

PHASE TRANSITION

• Imagine the !"# are drawn independently and

uniformly at random from [0,1]

• Question 1: If ) = +/2, what is the

probability that an envy-free allocation exists?

1. 2.

2/+

3.

1/2

4.

1

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15780 Spring 2019: Lecture 2

PHASE TRANSITION

• Imagine the !"# are drawn independently and

uniformly at random from [0,1]

• Question 2: If ) ≫ +, what is the probability

that an envy-free allocation exists?

1.

Close to 0

2.

Close to 1/3

3.

Close to 1/2

4.

Close to 1

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15780 Spring 2019: Lecture 2

PHASE TRANSITION

• Imagine the !"# are drawn independently and

uniformly at random from [0,1]

• Question 2: If ) ≫ +, what is the probability

that an envy-free allocation exists?

1.

Close to 0

2.

Close to 1/3

3.

Close to 1/2

4.

Close to 1

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15780 Spring 2019: Lecture 2

SHARP TRANSITION

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[Dickerson et al., AAAI 2014]

15780 Spring 2019: Lecture 2

SHARP TRANSITION

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[Cheeseman et al., IJCAI 1993]

15780 Spring 2019: Lecture 2

APPLICATION: KIDNEY EXCHANGE

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Donor 2 Patient 2 Donor 1 Patient 1

15780 Spring 2019: Lecture 2

KIDNEY EXCHANGE

• CONSTRUCT DIRECTED GRAPH:

EACH NODE IS A (DONOR, PATIENT) PAIR

• Edge from one node to another

if donor of first can donate to patient of second

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UNOS pool, Dec 2010 [Courtesy John Dickerson]

15780 Spring 2019: Lecture 2

KIDNEY EXCHANGE

• CYCLE-COVER: Given a

directed graph ! and " ∈ ℕ, find a collection of disjoint cycles of length ≤ " in ! that maximizes the number of covered vertices

• The problem is:
• Easy for " = 2 (why?)
• Easy for unbounded "
• NP-hard for a constant

" ≥ 3

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15780 Spring 2019: Lecture 2

KIDNEY EXCHANGE

• Variables: For each cycle ! of length

ℓ# ≤ %, variable &# ∈ {0,1}, &# = 1 iff cycle ! is included in the cover

• CYCLE-COVER as an IP:

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max ∑# &#ℓ# s.t. ∀6 ∈ 7, ∀!, &# ∈ {0,1} ∑#:9∈# &# ≤ 1

15780 Spring 2019: Lecture 2

SUMMARY

• IP tricks:
• TUM
• Binary variables
• Max min and min max
• Big ideas:
• IP representation can lead to “efficient” solutions
• Can prove theoretical guarantees by w.l.o.g.

relaxation to LP in some cases

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