Graduate AI Lecture 8: Integer Programming Applications Instructors: Nihar B. Shah (this time) J. Zico Kolter
A PPLICATION : R EVIEWER ASSIGNMENT IN P EER R EVIEW • 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 # $% ∈ [(, *] o Higher similarity score ⇒ Better envisaged quality of review o 15780 Spring 2019: Lecture 2 2
M AXIMIZING S UM S IMILARITIES • Assignment in many conferences aims to optimize the “sum similarity” • Maximize sum of similarities of all assigned (reviewer, paper) pairs 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 Nihar B. Shah, CMU 3
M AXIMIZING S UM S IMILARITIES maximize ' ' 5 (0 6 paper i assigned to reviewer j ( ∈ *+,-./ 0 ∈1-23-4-./ subject to every paper gets at least λ reviewers every reviewer gets at most μ papers 15780 Spring 2019: Lecture 2 Nihar B. Shah, CMU 4
I NTEGER P ROGRAM maximize - - 4 .2 5 .2 ' ∈) * × , . ∈[0] 2 ∈[3] subject to ∑ .∈[3] 5 .2 ≤ 8 ∀ : ∑ . ∈[3] 5 .2 ≥ < ∀ = 5 .2 ∈ 0,1 ∀ =, : 15780 Spring 2019: Lecture 2 Nihar B. Shah, CMU 5
T OTALLY U NIMODULAR M ATRICES 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 Nihar B. Shah, CMU 6
T OTALLY U NIMODULAR M ATRICES Definition: A matrix is called a Totally Unimodular Matrix (TUM) if every square submatrix has a determinant -1, 0, or 1. 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 i th column of A with b. 15780 Spring 2019: Lecture 2 Nihar B. Shah, CMU 7
H OMEWORK Is the constraint in the following problem TUM? maximize - - 4 .2 5 .2 ' ∈) * × , . ∈[0] 2 ∈[3] subject to ∑ .∈[3] 5 .2 ≤ 8 ∀ : ∑ . ∈[3] 5 .2 ≥ < ∀ = 5 .2 ∈ 0,1 ∀ =, : 15780 Spring 2019: Lecture 2 8
B ACK TO R EVIEWER A SSIGNMENT A concern with current popular approach: (lack of) Fairness Suppose ! = 1, % = 2 Similarities: 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 • 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 9
A M ORE F AIR A PPROACH maximize ' ' 5 (0 6 paper i assigned to reviewer j ( ∈ *+,-./ 0 ∈1-23-4-./ maximize min ' 5 (0 6 paper i assigned to reviewer j ( ∈ *+,-./ 0 ∈1-23-4-./ [Stelmakh et al. 2019] 15780 Spring 2019: Lecture 2 Nihar B. Shah, CMU 10
A M ORE F AIR A PPROACH maximize ' ∈) * × , min . ∈[0] 2 5 .3 6 .3 3 ∈[4] 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 Nihar B. Shah, CMU 11
A M ORE F AIR A PPROACH maximize - ' ∈) * × , such that ∑ / ∈[1] 3 4/ 5 4/ ≥ - ∀ 8 ∑ 4∈[1] 5 4/ ≤ : ∀ ; ∑ 4 ∈[1] 5 4/ ≥ < ∀ 8 5 4/ ∈ 0,1 ∀ 8, ; 15780 Spring 2019: Lecture 2 Nihar B. Shah, CMU 12
A PPLICATION : S UDOKU 8 4 6 7 4 1 6 5 5 9 3 7 8 7 4 8 2 1 3 5 2 9 1 3 9 2 5 15780 Spring 2019: Lecture 2 13
S UDOKU +, s.t. ) * +, = 1 • For each !, #, $ ∈ [9] , binary variable ) * iff we put $ in entry (!, #) • For t = 1, … , 27 , 5 6 is a row, column, or 3×3 square >> , … , ) ? ?? find ) > +, = 1 ∑ +,, ∈H I ) * s.t. ∀C ∈ 27 , ∀$ ∈ [9], +, = 1 ∀!, # ∈ 9 , ∑ *∈[?] ) * +, ∈ {0,1} ∀!, #, $ ∈ [9], ) * 15780 Spring 2019: Lecture 2 14
Note: If you have a hard time expressing something as an IP, try using binary variables. 15780 Spring 2019: Lecture 2 15
A PPLICATION : E NVY -F REENESS • Players ! = {1, … , '} and items ) = {1, … , *} • Player + has value , -. for item / • Partition items to bundles 0 1 , … , 0 2 • 0 1 , … , 0 2 is envy-free iff ∀+, + 4 , ∑ .∈7 8 , -. ≥ ∑ .∈7 8: , -. 1 2 $30 $50 $2 $5 $5 $3 $5 1 $2 $10 $5 $20 $20 $3 $40 2 15780 Spring 2019: Lecture 2 16
E NVY -F REENESS • Variables: ! "# ∈ 0,1 , ! "# = 1 iff ) ∈ * " • E NVY -F REE as an IP: find ! // , … , ! 12 ∀7 ∈ 8, ∀7 9 ∈ 8, s.t. ∑ #∈> ? "# ! "# ≥ ∑ #∈> ? "# ! " A # ∑ "∈B ! "# = 1 ∀) ∈ :, ∀7 ∈ 8, ) ∈ :, ! "# ∈ {0,1} 15780 Spring 2019: Lecture 2 17
P HASE T RANSITION • 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? 0 1. 2/+ 2. 1/2 3. 1 4. 15780 Spring 2019: Lecture 2 18
P HASE T RANSITION • 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? 0 1. 2/+ 2. 1/2 3. 1 4. 15780 Spring 2019: Lecture 2 19
P HASE T RANSITION • 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? Close to 0 1. Close to 1/3 2. Close to 1/2 3. Close to 1 4. 15780 Spring 2019: Lecture 2 20
P HASE T RANSITION • 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? Close to 0 1. Close to 1/3 2. Close to 1/2 3. Close to 1 4. 15780 Spring 2019: Lecture 2 21
S HARP T RANSITION [Dickerson et al., AAAI 2014] 15780 Spring 2019: Lecture 2 22
S HARP T RANSITION [Cheeseman et al., IJCAI 1993] 15780 Spring 2019: Lecture 2 23
A PPLICATION : K IDNEY E XCHANGE Donor 1 Donor 2 Patient 1 Patient 2 15780 Spring 2019: Lecture 2 24
K IDNEY E XCHANGE • C ONSTRUCT DIRECTED GRAPH : E ACH NODE IS A (D ONOR , P ATIENT ) PAIR Edge from one node to another • if donor of first can donate to patient of second UNOS pool, Dec 2010 [Courtesy John Dickerson] 15780 Spring 2019: Lecture 2 25
K IDNEY E XCHANGE • C YCLE -C OVER : 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?) o Easy for unbounded " o NP-hard for a constant o " ≥ 3 15780 Spring 2019: Lecture 2 26
K IDNEY E XCHANGE • Variables: For each cycle ! of length ℓ # ≤ % , variable & # ∈ {0,1} , & # = 1 iff cycle ! is included in the cover • C YCLE -C OVER as an IP: max ∑ # & # ℓ # ∑ #:9∈# & # ≤ 1 s.t. ∀6 ∈ 7, ∀!, & # ∈ {0,1} 15780 Spring 2019: Lecture 2 27
S UMMARY • IP tricks: TUM o Binary variables o Max min and min max o • Big ideas: IP representation can lead to “efficient” solutions o Can prove theoretical guarantees by w.l.o.g. o relaxation to LP in some cases 15780 Spring 2019: Lecture 2 28
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