Representative Problems

COS 320 - Algorithm Design

Representative Problems

Stable Matching (1.1)

matching med-school students to hospitals

Goal: given a set of preferences among hospitals and med-school students, design a self-reinforcing admissions process

Stable Assignment

Assignment with no unstable pairs

Unstable Pair

Hospital \(h\) and student \(s\) form an unstable pair if both:

• \(h\) prefers \(s\) to one of its admitted students
• \(s\) prefers \(h\) to assigned hospital

• Natural and desirable condition
• Individual self-interest prevents any hospital-student side deal

stable matching problem: input

Input: a set of \(n\) hospitals \(H\) and a set of \(n\) students \(S\) (one student per hospital for now)

• Each hospital \(h \in H\) ranks students
• Each student \(s \in S\) ranks hospitals

1st: most favorite; 3rd: least favorite

perfect matching

Matching

A matching \(M\) is a set of ordered pairs \(h\-s\) with \(h \in H\) and \(s \in S\) such that

• each hospital \(h \in H\) appears in at most one pair of \(M\), and
• each student \(s \in S\) appears in at most one pair of \(M\).

Perfect Matching

A matching \(M\) is perfect if \(|M| = |H| = |S| = n\)

perfect matching

A perfect matching \(M = \{ A\-Z, B\-Y, C\-X \}\)

unstable pair

Unstable Pair

Given a perfect matching \(M\), hospital \(h\) and student \(s\) form an unstable pair if both:

• \(h\) prefers \(s\) to matched student, and
• \(s\) prefers \(h\) to matched hospital

Key point: An unstable pair \(h\-s\) could each improve by joint action

unstable pair

\(A\-Y\) is an unstable pair for matching \(M = \{ A\-Z, B\-Y, C\-X \}\)

Here, Atlanta and Yolanda could make a side deal that would benefit both of them, but would leave Zeus and Boston in a bad position

stable matching problem

Stable Matching

A stable matching is a perfect matching with no unstable pairs.

Stable matching problem: given the preference lists of \(n\) hospitals and \(n\) students, find a stable matching (if one exists).

• Natural, desirable, and self-reinforcing condition
• Individual self-interest prevents any hospital-student pair from breaking commitment

stable matching problem

\(A\-Y\) is an unstable pair for matching \(M = \{ A\-Z, B\-Y, C\-X \}\)

A stable matching \(M = \{ A\-X, B\-Y, C\-Z \}\)

stable roommate problem

Q. Do stable matchings always exist?

1. Not obvious a priori.

Stable roommate problem:

• \(n\) rooms and \(2n\) people
• each person ranks others from \(1\) to \(2n-1\) (not self)
• Assign roommate pairs so that no unstable pairs exist

stable roommate problem

Q. Do stable matchings always exist?

1. Not obvious a priori.

Representative Problems

Stable Matching: gale-shapley

gale-shapley deferred acceptance alg

An intuitive method that guarantees to find a stable matching between \(n\) hospitals and \(n\) students

// Gale-Shapley (preference lists for hospitals and students)
Initialize M to empty matching
While some hospital h is unmatched and hasn't proposed to every student:
    s <- first student on h's list to whom h has not yet proposed
    If s is unmatched
        Add h-s to matching M
    Else If s prefers h to current partner h'
        Replace h'-s with h-s in matching M
    Else
        s rejects h
Return stable matching M

proof of correctness: termination

proof of correctness: termination

Claim: Gale-Shapley terminates after at most \(n^2\) iterations of while loop.

Pf: Each time through the while loop, a hospital proposes to a new student. There are only \(n^2\) possible proposals. ∎

Proposals required: \(n(n-1)+1\)

proof of correctness: perfection

Claim: Gale-Shapley produces a matching.

Pf:

• Hospitals proposes only if unmatched
  • each hospital matched to ≤1 student
• Student may accept a following offer, but only by rejecting previous offer.
  • student keeps only best hospital
  • each student matched to ≤1 hospital

Q. Do all hospitals and all students get matched?

proof of correctness: perfection

Claim: In Gale-Shapley matching, all hospitals get matched.

Pf: (by contradiction)

• Suppose, for sake of contradiction, that some hospital \(h \in H\) is not matched upon termination of Gale-Shapley algorithm.
• Then some student, say \(s \in S\), is not matched upon termination.
• By Observation 2, if \(s\) is not matched then \(s\) was never proposed to.
• But, \(h\) proposes to every student, since \(h\) ends up unmatched.

proof of correctness: perfection

Claim: In Gale-Shapley matching, all students get matched.

Pf:

• By previous claim, all \(n\) hospitals get matched.
• Thus, all \(n\) students get matched. ∎

proof of correctness: stability

Claim: In Gale-Shapley matching \(M'\), there are no unstable pairs.

Pf: Suppose that \(M'\) does not contain the pair \(h\)-\(s\).

• Case 1: \(h\) never proposed to \(s\).
  • \(h\) prefers its Gale-Shapley partner \(s'\) to \(s\)
    (hospitals propose in decreasing order of preference)
  • \(h\)-\(s\) is not unstable
• Case 2: \(h\) proposed to \(s\).
  • \(s\) rejected \(h\) (right away or later)
  • \(s\) prefers Gale-Shapley partner \(h'\) to \(h\)
    (students only trade up)
  • \(h\)-\(s\) is not unstable
• In either case, the pair \(h\)-\(s\) is not unstable. ∎

summary

Stable matching problem: Given \(n\) hospitals and \(n\) students and their preferences, find a stable matching if one exists.

Theorem [Gale-Shapley 1962]: The Gale-Shapley algorithm guarantees to find a stable matching for any problem instance.

Q. How to implement Gale-Shapley algorithm efficiently?

Q. If multiple stable matchings, which one does Gale-Shapley find?

Representative Problems

hospital optimality

understanding the solution

For a given problem instance, there may be several stable matchings.

An instance with two stable matchings:

\[M = \{ A\-X, B\-Y, C\-Z \}, \quad M' = \{ A\-Y, B\-X, C\-Z \}\]

understanding the solution

Valid Partner

Student \(s\) is a valid partner for hospital \(h\) if there exists any stable matching in which \(h\) and \(s\) are matched.

Ex:

• Both Xavier and Yolanda are valid partners for Atlanta.
• Both Xavier and Yolanda are valid partners for Boston.
• Zeus is the only valid partner for Chicago.

An instance with two stable matchings:

\[M = \{ A\-X, B\-Y, C\-Z \}, \quad M' = \{ A\-Y, B\-X, C\-Z \}\]

understanding the solution

Valid Partner

Student \(s\) is a valid partner for hospital \(h\) if there exists any stable matching in which \(h\) and \(s\) are matched.

Hospital-optimal assignment: Each hospital receives best valid partner.

• Is it perfect?
• Is it stable?

Claim: All executions of Gale-Shapley yield hospital-optimal assignment.

Corollary: Hospital-optimal assigment is a stable matching!

hospital optimality

Claim: Gale-Shapley matching \(M\) is hospital-optimal.

Proof: (by contradiction)

student pessimality

Q. Does hospital-optimality come at the expense of the students?

1. Yes.

Student-pessimal assignment: Each student receives worst valid partner.

Claim: Gale-Shapley finds student-pessimal stable matching \(M\)

Proof: (by contradiction)

Deceit: Machiavelli meets Gale-Shapley

Q. Can there be an incentive to misrepresent your preference list?

• Assume hospital's propose-and-reject algorithm is known
• Assume preference lists of all other participants are known

Fact: No, for any hospital; yes, for some students.

Example: Xavier😈 lies by swapping A ↔ C

Representative Problems

context

extensions

Extension 1: some agents declare others as unacceptable (med-school student unwilling to work in Cleveland)

Extension 2: some hospitals have more than one position.

Extension 3: Unequal number of positions and students (more than 43k med-school students; only 31k positions)

Matching \(M\) is unstable if there is a hospital \(h\) and student \(s\) such that:

• \(h\) and \(s\) are acceptable to each other; and
• either \(s\) is unmatched or \(s\) prefers \(h\) to assigned hospital; and
• either \(h\) does not have all its places filled or \(h\) prefers \(s\) to at least one of its assigned students.

extensions

Matching \(M\) is unstable if there is a hospital \(h\) and student \(s\) such that:

• \(h\) and \(s\) are acceptable to each other; and
• either \(s\) is unmatched or \(s\) prefers \(h\) to assigned hospital; and
• either \(h\) does not have all its places filled or \(h\) prefers \(s\) to at least one of its assigned students.

Theorem: There exists a stable matching.

Pf: Straightforward generalization of Gale-Shapley algorithm.

historical context

National Resident Matching Program (NRMP)

• Centralized clearinghouse to match med-school students to hospitals
• Began in 1952 to fix unraveling of offer dates (hospitals began making offers earlier and earlier, up to 2 years in advance)
• Originally used the "Boston Pool" algorithm
• Algorithm overhauled in 1998
  • med-school student optimal
  • deals with various side constraints (e.g., allow couples to match together... stable matching is no longer guaranteed to exist)

2012 Nobel Prize in Economics

Lloyd Shapley: Stable matching theory and Gale-Shapley algorithm.

Alvin Roth: Applied Gale-Shapley to matching med-school students with hospitals, students with schools, and organ donors with patients.

Other applications

New York City High School Matching

Other applications

QuestBridge National College Match for Low-Income Students

Other applications

Content delivery networks: Distribute much of world's content on web

Representative Problems

five representative problems (1.2)

Interval scheduling

Input: Set of jobs with start times and finish times.

Goal: Find maximum cardinatily (max count) subset of mutually compatible jobs (jobs don't overlap)

           a            :   :   :   :   :   :   
:        b      :   :   :   :   :   :   :   :   
:   :   :      c    :   :   :   :   :   :   :   
:   :   :            d          :   :   :   :   
:   :   :   :        e      :   :   :   :   :   
:   :   :   :   :          f        :   :   :   
:   :   :   :   :   :          g        :   :   
:   :   :   :   :   :   :   :        h      :   
:   :   :   :   :   :   :   :   :   :   :   :   
0   1   2   3   4   5   6   7   8   9   10  11  

Weighted interval scheduling

Input: Set of jobs with start times, finish times, and weights.

Goal: Find maximum weight subset of mutually compatible jobs (jobs don't overlap)

           23           :   :   :   :   :   :   
:        12     :   :   :   :   :   :   :   :   
:   :   :      20   :   :   :   :   :   :   :   
:   :   :            26         :   :   :   :   
:   :   :   :        13     :   :   :   :   :   
:   :   :   :   :          20       :   :   :   
:   :   :   :   :   :          11       :   :   
:   :   :   :   :   :   :   :        16     :   
:   :   :   :   :   :   :   :   :   :   :   :   
0   1   2   3   4   5   6   7   8   9   10  11  

bipartite matching

Problem: Given a bipartite graph \(G = (L \cup R, E)\), find a max cardinality matching.

A subset of edges \(M \subseteq E\) is a matching if each node appears in at most one edge in \(M\)

independent set

Problem: Given a graph \(G = (V, E)\), find a max cardinality independent set.

A subset \(S \subseteq V\) is independent if for every \((u,v) \in E\), either \(u \notin S\) or \(v \notin S\), or both.

competitive facility location

Input: Graph with weight on each node

Game: Two competing players alternate in selecting nodes. Not allowed to select a node if any of its neighbors have been selected.

Goal: Select a maximum weight subset of nodes.

five representative problems

Variations on a theme: independent set

• Interval Scheduling

  • \(O(n \log n)\) greedy algorithm

• Weighted Interval Scheduling

  • \(O(n \log n)\) dynamic programming algorithm

• Bipartite Matching

  • \(O(n^k)\) max-flow based algorithm

• Independent Set

  • NP-Complete

• Competitive Facility Location

  • PSPACE-complete

representative problems

efficient implementation of gale-shapley

efficient implementation

how to implement Gale-Shapley efficiently (\(O(n^2)\))

// Gale-Shapley (preference lists for hospitals and students)
Initialize M to empty matching
While some hospital h is unmatched and hasn't proposed to every student:
    s <- first student on h's list to whom h has not yet proposed
    If s is unmatched
        Add h-s to matching M
    Else If s prefers h to current partner h'
        Replace h'-s with h-s in matching M
    Else
        s rejects h
Return stable matching M

efficient implementation

Representing hospitals and students:

• Index hospitals and students \(1, \ldots, n\).

Representing the matching:

• Maintain a list of free hospitals (in a stack or queue)
• Maintain two arrays student[h] and hospital[s].
  • If \(h\) matched to \(s\), then student[h] = s and hospitals[s] = h
  • Use value \(0\) to designate that hospital or student is unmatched.

efficient implementation

Hospitals proposing:

• For each hospital, maintain a list of students, ordered by preference.
• For each hospital, maintain a pointer to students in list for next proposal.

efficient implementation

Students rejecting/accepting:

• Does student \(s\) prefer hospital \(h\) to hospital \(h'\)?
• For each student, create inverse of preference list of hospitals.
• Constant time access for each query after \(O(n)\) preprocessing.

for i = 1 to n
    inverse[pref[i]] = i

Student prefers hospital 3 to 6 since inverse[3] < inverse[6]

representative problems

gale-shapley practice