Popular Matching

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Popular Matching
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Example Instance

• Applicants
• Posts
• Preference lists
• Matching
• Better Matching

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Popular Matching

• Gar75 M2 more popular than M1 if
• applicants who prefer M2 to M1
• outnumber
• applicants who prefer M1 to M2
• M1 popular if no M2 more popular than M1

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Example Instance
M2
M1
M3

• Relation not transitive, or even acyclic
• Popular matching may not exist

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The Challenge

• Decide if given matching is popular
• Find popular matching, or determine no such
  matching exists

Aim efficiently checkable characterization
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Conventions

• Append dummy post to each list
• a1 p1 p2 p3 d1
• Matchings are applicant-complete
• M(a) is partner of a in M
• f(a) is as first-choice post
• pf(a) called an f-post
• f(p) is set of applicants who rank p first

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Necessary Conditions

• In any popular matching,
• Every f-post p must be matched
• And to an applicant in f(p)

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More Conventions

• s(a) is as first non-f-post
• Must exist s(a) can be dummy post
• ps(a)called an s-post
• Note f-posts disjoint from s-posts

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More Necessary Conditions

• In any popular matching,
• M(a) never strictly between f(a) and s(a)
• M(a) never worse than s(a)

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Efficient Characterization

• M is a popular matching iff
• Every f-post is matched
• For each applicant a, M(a) is f(a) or s(a)
• Perform self-reduction ? a f(a) s(a)
• M popular in reduced instance iff
• Every f-post matched
• M is applicant-complete

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Negative Example
reduction
Maximum matching size is 2 No applicant-complete
matching (Halls Marriage Theorem)
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Positive Example
reduction

• Matching is popular, since
• applicant-complete in reduced instance, and
• every f-post is matched

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Maximum Matching

• Reduced instance applicant degree is 2
• Hopcroft-Karp bound gives O(n3/2)
• O(n) time algorithm
• Phase 1
• Repeat
• Match degree-1 post to adjacent applicant
• Remove matched vertices any degree-0 posts

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Maximum Matching Continued

• P remaining posts (all have degree 2)
• A remaining applicants (all have degree 2)
• If A gt P then no applicant-complete matching
• By Halls Marriage Theorem
• Else, graph decomposes into disjoint cycles
• A P and deg(a) deg(p)
• But, 2A deg-sum of posts 2P
• Hence, A P and all posts have degree 2

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Further Related Results

• Max-card popular matching in linear time
• Preference lists with ties
• Generalize defn of f-post, s-post
• Optimal O(mvn) time algorithm
• Multi-capacity posts
• Voting Paths

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Voting Paths

• Non-popular matching M1 already in place
• May be no popular matching that is more popular
  than M1
• Voting path is a sequence of matchings
• ltM1,M2,M3,,Mk-1,Mkgt
• Mi more popular than Mi-1
• Mk popular

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Voting Path Problems

• For a given matching M1
• Is there a voting path beginning at M1?
• more popular than relation is not acyclic
• If so, what is the length of a shortest such
  path?
• there are an exponential number of matchings
• How efficiently can we solve these problems?

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Voting Path Results

• For a given matching M1
• There is always a voting path beginning at M1
• A shortest such path has length at most 3
• e.g. ltM1,M2,M3,M4gt
• Find such a path in O(mvn) time

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Definitions of Good

• Pareto optimal
• no other matching in which some applicant is
  better off, and no applicant is worse off
• Max-utility
• k points for first choice, (k 1) for second,
• maximum-weight matching
• Rank-maximal
• allocate maximum number to 1st choice, and
  subject to this, allocate
  maximum number to 2nd choice, and so on
• Fair
• allocate minimum to last choice, then second
  last