Popular Matching David Abraham Carnegie Mellon University Kavitha Telikepalli Max-Planck-Institut für Informatik Kurt Mehlhorn Rob Irving University of Glasgow

Popular Matching - David Abraham Example Instance Applicants Posts Preference lists Matching Better Matching 1234 a1a1 p1p1 p4p4 a2a2 p 2 p 3 p1p1 a3a3 p2p2 p3p3 a4a4 p1p1 p2p2 p3p3 p4p4

Popular Matching - David Abraham Popular Matching [Gar75] M 2 more popular than M 1 if applicants who prefer M 2 to M 1 outnumber applicants who prefer M 1 to M 2 M 1 popular if no M 2 more popular than M 1

Example Instance 123 a1a1 p1p1 p2p2 p3p3 a2a2 p1p1 p2p2 p3p3 a3a3 p1p1 p2p2 p3p3 M1M1 M2M2 M3M3 Relation not transitive, or even acyclic Popular matching may not exist

Popular Matching - David Abraham The Challenge 1.Decide if given matching is popular 2.Find popular matching, or determine no such matching exists Aim: efficiently checkable characterization

Popular Matching - David Abraham Conventions Append dummy post to each list a 1 : p 1 p 2 p 3 d 1 –Matchings are applicant-complete –M(a) is partner of a in M f(a) is a ’s first-choice post –p=f(a) called an f-post –f(p) is set of applicants who rank p first

Popular Matching - David Abraham Necessary Conditions In any popular matching, … 1.Every f-post p must be matched 2.And to an applicant in f(p) 123 a1a1 p1p1 p2p2 d1d1 a2a2 p1p1 p3p3 d2d2 a3a3 p3p3 p4p4 d3d3

Popular Matching - David Abraham More Conventions s(a) is a ’s first non-f-post –Must exist: s(a) can be dummy post –p=s(a)called an s-post Note: f-posts disjoint from s-posts

Popular Matching - David Abraham More Necessary Conditions In any popular matching,… 3.M(a) never strictly between f(a) and s(a) 4.M(a) never worse than s(a) 1234 a1a1 p1p1 p3p3 p2p2 d1d1 a2a2 p1p1 p3p3 p2p2 d2d2 a3a3 p1p1 p3p3 p4p4 d3d3 a4a4 p3p3 p1p1 p4p4 d4d4

Popular Matching - David Abraham Efficient Characterization M is a popular matching iff i.Every f-post is matched ii.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 i. Every f-post matched ii. M is applicant-complete

Popular Matching - David Abraham Negative Example 1234 a1a1 p1p1 p2p2 p3p3 d1d1 a2a2 p1p1 p2p2 p3p3 d2d2 a3a3 p1p1 p2p2 p3p3 d3d3 fs a1a1 p1p1 p2p2 a2a2 p1p1 p2p2 a3a3 p1p1 p2p2 Maximum matching size is 2 No applicant-complete matching (Hall’s Marriage Theorem) reduction

Popular Matching - David Abraham Positive Example 1234 a1a1 p1p1 p2p2 d1d1 a2a2 p1p1 p3p3 p2p2 d2d2 a3a3 p3p3 p2p2 d3d3 a4a4 p3p3 p1p1 d4d4 fs a1a1 p1p1 p2p2 a2a2 p1p1 p2p2 a3a3 p3p3 p2p2 a4a4 p3p3 d4d4 reduction Matching is popular, since i. applicant-complete in reduced instance, and ii. every f-post is matched

Popular Matching - David Abraham Maximum Matching Reduced instance: applicant degree is 2 Hopcroft-Karp bound gives O(n 3/2 ) O(n) time algorithm Phase 1: –Repeat Match degree-1 post to adjacent applicant Remove matched vertices & any degree-0 posts

Popular Matching - David Abraham Maximum Matching Continued P = remaining posts (all have degree ≥ 2) A = remaining applicants (all have degree = 2) If |A| > |P| then no applicant-complete matching –By Hall’s Marriage Theorem Else, graph decomposes into disjoint cycles –|A| ≤ |P| and deg(a) ≤ deg(p) –But, 2|A| = deg-sum of posts ≥ 2|P| –Hence, |A| = |P| and all posts have degree = 2

Popular Matching - David Abraham Further Related Results Max-card popular matching in linear time Preference lists with ties –Generalize defn of f-post, s-post –Optimal O(m√n) time algorithm Multi-capacity posts Voting Paths

Popular Matching - David Abraham Voting Paths Non-popular matching M 1 already in place May be no popular matching that is more popular than M 1 Voting path is a sequence of matchings – –M i more popular than M i-1 –M k popular

Popular Matching - David Abraham Voting Path Problems For a given matching M 1 : –Is there a voting path beginning at M 1 ? “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?

Popular Matching - David Abraham Voting Path Results For a given matching M 1 : –There is always a voting path beginning at M 1 –A shortest such path has length at most 3 e.g. –Find such a path in O(m√n) time

Popular Matching - David Abraham 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 1 st choice, and subject to this, allocate maximum number to 2 nd choice, and so on Fair: –allocate minimum to last choice, then second last…