Some Representative Problems (1)

COS 320 - Algorithm Design

Some Representative Problems (1)

Stable Matching

matching med-school students to hospitals

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

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

Stable Assignment

Assigment with no unstable pairs

• 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

quiz 0: unstable pair

Which pair is unstable in matching \(M = \{ A\-X, B\-Z, C\-Y \}\)?

1. \(A\-Y\)

2. \(B\-X\)

3. \(B\-Z\)

4. None of the above

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 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:

• \(2n\) people; each person ranks others from \(1\) to \(2n-1\)
• Assign roommate pairs so that no unstable pairs exist

stable roommate problem

Q. Do stable matchings always exist?

1. Not obvious a priori.

Observation: Stable matchings need not exist

Some Representative Problems (1)

Stable Matching: gale-shapley

gale-shapley deferred acceptance alg

An intuitive method that guarantees to find a stable matching

// 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

Observation 1: Hospitals propose to students in decreasing order of preference.

Observation 2: Once a student is matched, the student never becomes unmatched; only "trades up"

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

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, \(s\) was never proposed to.
• But, \(h\) proposes to every student, since \(h\) ends up unmatched.

“

Observation 2: Once a student is matched, the student never becomes unmatched; only "trades up"

”

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?

quiz 1: same stable matching?

Do all executions of Gale-Shapley lead to the same stable matching?

1. No, because the algorithm is nondeterministic

2. No, because an instance can have several stable matchings

3. Yes, because each instance has a unique stable matching

4. Yes, even though an instance can have several stable matchings and the algorithm is nondeterministic

Some Representative Problems (1)

hospital optimality

understanding the solution

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

An instance with two stable matchings:

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

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:

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

quiz 2: valid partner

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?
A: Yes.

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

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

Proof: (by contradiction)

quiz 3: lie for gain?

Suppose each agent knows the preference lists of every other agent before the hospital propose-and-reject algorithm is executed. Which of the following is true?

1. No hospital can improve by falsifying its preference list

2. No student can improve by falsifying their preference list

3. Both A and B

4. Neither A nor B

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

Some Representative Problems (1)

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

Some Representative Problems (1)

five representative problems

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)

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)

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

some representative problems (1)

efficient implementation of gale-shapley

efficient implementation

Efficient implementation: We describe an \(O(n^2)\) time 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]

Some representative problems (1)

gale-shapley practice

practice