[PPT] - Classified Matchings under one sided preferences Meghana Nasre IIT PowerPoint Presentation

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Classified Matchings under one sided preferences

Meghana Nasre IIT Madras Recent Trends in Algorithms NISER, Bhubaneshwar

Feb 07, 2019 joint work with Prajakta Nimbhorkar (CMI) and Nada Pulath (IIT-M)

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Classified Matchings under one sided preferences

Problem setup

Input: A set of applicants A. A set of posts P. Applicants have preferences over a subset in P. a1 a2 a3 p1 p2 1 2 2 1 2 1

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Classified Matchings under one sided preferences

Problem setup

Input: A set of applicants A. A set of posts P. Applicants have preferences over a subset in P. Posts have quotas. a1 a2 a3 p1 p2 1 2 2 1 2 1 (2) (2)

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Classified Matchings under one sided preferences

Problem setup

Input: A set of applicants A. A set of posts P. Applicants have preferences over a subset in P. Posts have quotas. Posts have classes. a1 a2 a3 p1 p2 1 2 2 1 2 1 (2) (2) 1 1

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Classified Matchings under one sided preferences

Problem setup

Input: A set of applicants A. A set of posts P. Applicants have preferences over a subset in P. Posts have quotas. Posts have classes. a1 a2 a3 p1 p2 1 2 2 1 2 1 (2) (2) 1 1 Classes are subsets on the neighborhood.

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Classified Matchings under one sided preferences

Problem setup

Input: A set of applicants A. A set of posts P. Applicants have preferences over a subset in P. Posts have quotas. Posts have classes. a1 a2 a3 p1 p2 1 2 2 1 2 1 1 1 2 2 The neighborhood is a trivial class!

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Classified Matchings under one sided preferences

Problem setup

Input: A set of applicants A. A set of posts P. Applicants have preferences over a subset in P. Posts have quotas. Posts have classes. a1 a2 a3 p1 p2 1 2 2 1 2 1 1 1 2 2 Goal: Match applicants to posts optimally.

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Classified Matchings under one sided preferences

Why classifications?

Some natural constraints that can be modelled: Allotting courses to students Course - may not want many students from the same Dept. Allotting tasks to employees Task - wasteful to have many employees with the same skills.

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Classified Matchings under one sided preferences

Problem Setup

Input: A set of applicants A. A set of posts P. Applicants have preferences over a subset in P. Posts have quotas. Posts have classes. a1 a2 a3 p1 p2 1 1 2 2 Goal: Compute a maximum cardinality matching.

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Classified Matchings under one sided preferences

Problem Setup

Input: A set of applicants A. A set of posts P. Applicants have preferences over a subset in P. Posts have quotas. Posts have classes. a1 a2 a3 p1 p2 1 1 2 2 Goal: Compute a maximum cardinality matching. Arbitrary classes then problem is NP-Hard.

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Classified Matchings under one sided preferences

Problem Setup

Input: A set of applicants A. A set of posts P. Applicants have preferences over a subset in P. Posts have quotas. Posts have classes. a1 a2 a3 p1 p2 1 1 2 2 Goal: Compute a maximum cardinality matching. Arbitrary classes then problem is NP-Hard. Consider special classification families.

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Classified Matchings under one sided preferences

Laminar classification

Huang (2010); 2-sided pref.

Laminar classification ⇐ ⇒ any pair of classes is non-intersecting OR Example: Countries , States , Districts , Cities Special case: Partition

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Classified Matchings under one sided preferences

Problem setup

Input: A set of applicants A A set of posts P Applicants have preferences over a subset in P Posts have quotas Posts have laminar classes and quotas a1 a2 a3 p1 p2 1 1 2 2 Goal: Compute a maximum cardinality matching.

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Classified Matchings under one sided preferences

Maximum matchings under laminar classifications

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Classified Matchings under one sided preferences

Maximum matchings under laminar classifications ⇓ Maximum flow in a flow network

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Classified Matchings under one sided preferences

Classification tree - property of laminar classification

a1 a3 a2 p1(2) p2(2) 1 1 C1 C2

a1 a3 a2 p2a2 p2a3 p2a1 p1a1 p1a3 p1a2 p1 p2 C1 C2

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Classified Matchings under one sided preferences

Feasible matchings to feasible flows

a1 a3 a2 p1(2) p2(2) 1 1 C1 C2

a1 a3 a2 s p2a2 p2a3 p2a1 p1a1 p1a3 p1a2 p1 p2 t C1 C2 2 2

G H Add source s and sink t

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Classified Matchings under one sided preferences

Feasible matchings to feasible flows

a1 a3 a2 p1(2) p2(2) 1 1 C1 C2

a1 a3 a2 s p2a2 p2a3 p2a1 p1a1 p1a3 p1a2 p1 p2 t C1 C2 2 2

G H Add (a, pa) edges

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Classified Matchings under one sided preferences

Feasible matchings to feasible flows

a1 a3 a2 p1(2) p2(2) 1 1 C1 C2

a1 a3 a2 s p2a2 p2a3 p2a1 p1a1 p1a3 p1a2 p1 p2 t C1 C2 2 2

G H Compute a max-flow in H

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Classified Matchings under one sided preferences

Feasible matchings to feasible flows

a1 a3 a2 p1(2) p2(2) 1 1 C1 C2

a1 a3 a2 s p2a2 p2a3 p2a1 p1a1 p1a3 p1a2 p1 p2 t C1 C2 2

G H Maximum matching in G ⇐ ⇒ Maximum flow in H

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Classified Matchings under one sided preferences

Back to our problem

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Classified Matchings under one sided preferences

Back to our problem

Input: A set of applicants A. A set of posts P. Applicants have preferences over a subset in P. Posts have quotas. Posts have laminar classes. a1 a3 a2 p1 p2 1 2 2 1 2 1 1 1 2 2 Goal: Match applicants to posts optimally.

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Classified Matchings under one sided preferences

Back to our problem

Input: A set of applicants A. A set of posts P. Applicants have preferences over a subset in P. Posts have quotas. Posts have laminar classes. a1 a3 a2 p1 p2 1 2 2 1 2 1 1 1 2 2 Goal: Match applicants to posts optimally. Popularity: majority does not want to deviate.

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Classified Matchings under one sided preferences

Back to our problem

Input: A set of applicants A. A set of posts P. Applicants have preferences over a subset in P. Posts have quotas. Posts have laminar classes. a1 a3 a2 p1 p2 1 2 2 1 2 1 1 1 2 2 Goal: Match applicants to posts optimally. Rank-maximality: max. number to rank-1, subject to this max. number to rank-2, ...

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Classified Matchings under one sided preferences

Popularity in the one-to-one setting

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Classified Matchings under one sided preferences

Popularity: definition

G¨ ardenfors (1975), Abraham et al.(2005)

Compare matchings M and M′ using applicant votes.

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Classified Matchings under one sided preferences

Popularity: definition

G¨ ardenfors (1975), Abraham et al.(2005)

Compare matchings M and M′ using applicant votes. a1 a2 a3 p1 p2 p3 1 1 1 1 1 2 2 a1 a2 a3 p1 p2 p3 1 1 1 1 1 2 2

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Classified Matchings under one sided preferences

Popularity: definition

G¨ ardenfors (1975), Abraham et al.(2005)

Compare matchings M and M′ using applicant votes. a1 a2 a3 p1 p2 p3 1 1 1 1 1 2 2 a1 a2 a3 p1 p2 p3 1 1 1 1 1 2 2 M M′ a1

a2
a3

✓

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Classified Matchings under one sided preferences

Popularity: definition

G¨ ardenfors (1975), Abraham et al.(2005)

Compare matchings M and M′ using applicant votes. a1 a2 a3 p1 p2 p3 1 1 1 1 1 2 2 a1 a2 a3 p1 p2 p3 1 1 1 1 1 2 2 M M′ a1

a2
a3

✓

M beats M′ =

⇒ M′ is not popular

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Classified Matchings under one sided preferences

Popularity: definition

G¨ ardenfors (1975), Abraham et al.(2005)

Compare matchings M and M′ using applicant votes. a1 a2 a3 p1 p2 p3 1 1 1 1 1 2 2 a1 a2 a3 p1 p2 p3 1 1 1 1 1 2 2 M M′ a1

a2
a3

✓

M beats M′ =

⇒ M′ is not popular A matching M is popular if no matching beats it.

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Classified Matchings under one sided preferences

Popularity: definition

G¨ ardenfors (1975), Abraham et al.(2005)

Compare matchings M and M′ using applicant votes. a1 a2 a3 p1 p2 p3 1 1 1 1 1 2 2 a1 a2 a3 p1 p2 p3 1 1 1 1 1 2 2 M M′ a1

a2
a3

✓

M beats M′ =

⇒ M′ is not popular A matching M is popular if no matching beats it.

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Classified Matchings under one sided preferences

Popular matchings: characterization

Abraham et al. (2005)

A matching M is popular if no matching beats it.

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Classified Matchings under one sided preferences

Popular matchings: characterization

Abraham et al. (2005)

A matching M is popular if no matching beats it. A matching M is popular if and only if: M is a maximum matching on the rank-1 edges. Every a ∈ A is matched to either its f (a) or s(a)

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Classified Matchings under one sided preferences

Popular matchings: characterization

Abraham et al. (2005)

A matching M is popular if no matching beats it. A matching M is popular if and only if: M is a maximum matching on the rank-1 edges. Every a ∈ A is matched to either its f (a) or s(a) f (a) - set of all rank-1 posts of a. s(a) - next most preferred posts of a.

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Classified Matchings under one sided preferences

Popular matchings: characterization

Abraham et al. (2005)

A matching M is popular if no matching beats it. A matching M is popular if and only if: M is a maximum matching on the rank-1 edges. Every a ∈ A is matched to either its f (a) or s(a) f (a) - set of all rank-1 posts of a. s(a) - next most preferred posts of a. If such a matching does not exist, no popular matching exists.

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Classified Matchings under one sided preferences

Computing popular matchings

Abraham et al. (2005); Manlove and Sng (2006)

Overall idea: Reduction to two maximum matching computations.

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Classified Matchings under one sided preferences

Computing popular matchings

Abraham et al. (2005); Manlove and Sng (2006)

Overall idea: Reduction to two maximum matching computations.

1 Construct G1: every a adds (a, f (a)) edges.

G1 is graph on rank-1 edges.

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Classified Matchings under one sided preferences

Computing popular matchings

Abraham et al. (2005); Manlove and Sng (2006)

Overall idea: Reduction to two maximum matching computations.

1 Construct G1: every a adds (a, f (a)) edges.

G1 is graph on rank-1 edges.

2 Compute a maximum matching M1 in G1. 3 Delete “unnecessary” rank-1 edges.

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Classified Matchings under one sided preferences

Computing popular matchings

Abraham et al. (2005); Manlove and Sng (2006)

Overall idea: Reduction to two maximum matching computations.

1 Construct G1: every a adds (a, f (a)) edges.

G1 is graph on rank-1 edges.

2 Compute a maximum matching M1 in G1. 3 Delete “unnecessary” rank-1 edges. 4 Some applicants add (a, s(a)) edges.

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Classified Matchings under one sided preferences

Computing popular matchings

Abraham et al. (2005); Manlove and Sng (2006)

Overall idea: Reduction to two maximum matching computations.

1 Construct G1: every a adds (a, f (a)) edges.

G1 is graph on rank-1 edges.

2 Compute a maximum matching M1 in G1. 3 Delete “unnecessary” rank-1 edges. 4 Some applicants add (a, s(a)) edges. 5 Augment M1 to compute a maximum matching M.

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Classified Matchings under one sided preferences

Computing popular matchings

Abraham et al. (2005); Manlove and Sng (2006)

Overall idea: Reduction to two maximum matching computations.

1 Construct G1: every a adds (a, f (a)) edges.

G1 is graph on rank-1 edges.

2 Compute a maximum matching M1 in G1. 3 Delete “unnecessary” rank-1 edges. 4 Some applicants add (a, s(a)) edges. 5 Augment M1 to compute a maximum matching M. 6 If M matches all applicants, declare popular,

else no popular matching.

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Classified Matchings under one sided preferences

Computing popular matchings

Abraham et al. (2005); Manlove and Sng (2006)

Overall idea: Reduction to two maximum matching computations.

1 Construct G1: every a adds (a, f (a)) edges.

G1 is graph on rank-1 edges.

2 Compute a maximum matching M1 in G1. 3 Delete “unnecessary” rank-1 edges. 4 Some applicants add (a, s(a)) edges. 5 Augment M1 to compute a maximum matching M. 6 If M matches all applicants, declare popular,

else no popular matching. Steps 3 & 4: Dulmage Mendelsohn Decomposition

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Classified Matchings under one sided preferences

Dulmage Mendelsohn decomposition

DM (1958)

Partition of vertices into three sets w.r.t. a maximum matching.

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Classified Matchings under one sided preferences

Dulmage Mendelsohn decomposition

DM (1958)

Partition of vertices into three sets w.r.t. a maximum matching. a1 a2 a3 a4 p1 p2 p3 p4

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Classified Matchings under one sided preferences

Dulmage Mendelsohn decomposition

DM (1958)

Partition of vertices into three sets w.r.t. a maximum matching. a1 a2 a3 a4 p1 p2 p3 p4 M is a maximum matching

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Classified Matchings under one sided preferences

Dulmage Mendelsohn decomposition

DM (1958)

Partition of vertices into three sets w.r.t. a maximum matching. a1 a2 (E)a3 a4 p1 p2 p3(E) p4 M is a maximum matching E : reachable from unmatched vertex via even length alt. path.

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Classified Matchings under one sided preferences

Dulmage Mendelsohn decomposition

DM (1958)

Partition of vertices into three sets w.r.t. a maximum matching. a1 (O)a2 (E)a3 a4 p1(O) p2 p3(E) p4 M is a maximum matching E : reachable from unmatched vertex via even length alt. path. O : reachable from unmatched vertex via odd length alt. path.

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Classified Matchings under one sided preferences

Dulmage Mendelsohn decomposition

DM (1958)

Partition of vertices into three sets w.r.t. a maximum matching. (E)a1 (O)a2 (E)a3 a4 p1(O) p2(E) p3(E) p4 M is a maximum matching E : reachable from unmatched vertex via even length alt. path. O : reachable from unmatched vertex via odd length alt. path.

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Classified Matchings under one sided preferences

Dulmage Mendelsohn decomposition

DM (1958)

Partition of vertices into three sets w.r.t. a maximum matching. (E)a1 (O)a2 (E)a3 a4 p1(O) p2(E) p3(E) p4 M is a maximum matching E : reachable from unmatched vertex via even length alt. path. O : reachable from unmatched vertex via odd length alt. path.

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Classified Matchings under one sided preferences

Dulmage Mendelsohn decomposition

DM (1958)

Partition of vertices into three sets w.r.t. a maximum matching. (E)a1 (O)a2 (E)a3 (U)a4 p1(O) p2(E) p3(E) p4(U) M is a maximum matching E : reachable from unmatched vertex via even length alt. path. O : reachable from unmatched vertex via odd length alt. path. U : unreachable from unmatched vertex via length alt. path.

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SLIDE 51

Classified Matchings under one sided preferences

Dulmage Mendelsohn decomposition

DM (1958)

Partition of vertices into three sets w.r.t. a maximum matching. (E)a1 (O)a2 (E)a3 (U)a4 p1(O) p2(E) p3(E) p4(U)

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Classified Matchings under one sided preferences

Dulmage Mendelsohn decomposition

DM (1958)

Partition of vertices into three sets w.r.t. a maximum matching. (E)a1 (O)a2 (E)a3 (U)a4 p1(O) p2(E) p3(E) p4(U) E, O and U invariant of M

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SLIDE 53

Classified Matchings under one sided preferences

Dulmage Mendelsohn decomposition

DM (1958)

Partition of vertices into three sets w.r.t. a maximum matching. (E)a1 (O)a2 (E)a3 (U)a4 p1(O) p2(E) p3(E) p4(U) E, O and U invariant of M For any maximum matching:

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SLIDE 54

Classified Matchings under one sided preferences

Dulmage Mendelsohn decomposition

DM (1958)

Partition of vertices into three sets w.r.t. a maximum matching. (E)a1 (O)a2 (E)a3 (U)a4 p1(O) p2(E) p3(E) p4(U) E, O and U invariant of M For any maximum matching: Every O and U vertex is matched.

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Classified Matchings under one sided preferences

Dulmage Mendelsohn decomposition

DM (1958)

Partition of vertices into three sets w.r.t. a maximum matching. (E)a1 (O)a2 (E)a3 (U)a4 p1(O) p2(E) p3(E) p4(U) E, O and U invariant of M For any maximum matching: Every O and U vertex is matched. No OO,OU edges are matched.

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SLIDE 56

Classified Matchings under one sided preferences

Computing popular matchings

Abraham et al. (2005); Manlove and Sng (2006)

Overall idea: Reduction to two maximum matching computations.

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SLIDE 57

Classified Matchings under one sided preferences

Computing popular matchings

Abraham et al. (2005); Manlove and Sng (2006)

Overall idea: Reduction to two maximum matching computations.

1 Construct G1: every a adds (a, f (a)) edges.

G1 is graph on rank-1 edges.

2 Compute a maximum matching M1 in G1. 3 Delete “unnecessary” rank-1 edges. OO,OU edges. 4 Even applicants add (a, s(a)) edges. 5 Augment M1 to compute a maximum matching M. 6 If M matches all applicants, declare popular,

else no popular matching.

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SLIDE 58

Classified Matchings under one sided preferences

Computing popular matchings

Abraham et al. (2005); Manlove and Sng (2006)

Overall idea: Reduction to two maximum matching computations.

1 Construct G1: every a adds (a, f (a)) edges.

G1 is graph on rank-1 edges.

2 Compute a maximum matching M1 in G1. 3 Delete “unnecessary” rank-1 edges. OO,OU edges. 4 Even applicants add (a, s(a)) edges. 5 Augment M1 to compute a maximum matching M. 6 If M matches all applicants, declare popular,

else no popular matching. Step 4: s(a) – most preferred even post in G1.

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Classified Matchings under one sided preferences

Is deletion necessary?

a1 a2 a3 p1 p2 p3 1 2 1 1 1 1 2

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SLIDE 60

Classified Matchings under one sided preferences

Is deletion necessary?

a1 a2 a3 p1 p2 p3 1 2 1 1 1 1 2

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SLIDE 61

Classified Matchings under one sided preferences

Is deletion necessary?

a1 a2 a3 p1 p2 p3 1 2 1 1 1 1 2 a1 a2 a3 p1 p2 p3 1 1 1 1 1 Graph G1

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Classified Matchings under one sided preferences

Is deletion necessary?

a1 a2 a3 p1 p2 p3 1 2 1 1 1 1 2 a1 a2 a3 p1 p2 p3 1 1 1 1

Max. matching M1 in G1

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Classified Matchings under one sided preferences

Is deletion necessary?

a1 a2 a3 p1 p2 p3 1 2 1 1 1 1 2 (E)a1 (O)a2 (E)a3 p1(O) p2(E) p3(E) 1 1 1 1 Label vertices

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Classified Matchings under one sided preferences

Is deletion necessary?

a1 a2 a3 p1 p2 p3 1 2 1 1 1 1 2 (E)a1 (O)a2 (E)a3 p1(O) p2(E) p3(E) 1 1 1 1 2 2 Add (a, s(a)) edges

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SLIDE 65

Classified Matchings under one sided preferences

Is deletion necessary?

a1 a2 a3 p1 p2 p3 1 2 1 1 1 1 2 (E)a1 (O)a2 (E)a3 p1(O) p2(E) p3(E) 1 1 1 1 2 2 Augment M1 to get M′

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Classified Matchings under one sided preferences

Is deletion necessary?

a1 a2 a3 p1 p2 p3 1 2 1 1 1 1 2 (E)a1 (O)a2 (E)a3 p1(O) p2(E) p3(E) 1 1 1 1 2 2 M M′ a1

a2
a3

✓

M′ is not popular

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SLIDE 67

Classified Matchings under one sided preferences

Is deletion necessary?

a1 a2 a3 p1 p2 p3 1 2 1 1 1 1 2 (E)a1 (O)a2 (E)a3 p1(O) p2(E) p3(E) 1 1 1 1 2 2 Deletion of OO, OU edges is crucial!

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SLIDE 68

Classified Matchings under one sided preferences

Back to our problem

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SLIDE 69

Classified Matchings under one sided preferences

Back to our problem

Input: A set of applicants A. A set of posts P. Applicants have preferences over a subset in P. Posts have quotas. Posts have laminar classes. a1 a3 a2 p1 p2 1 2 2 1 2 1 1 1 2 2 Goal: Compute a popular matching of applicants to posts (if one exists).

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SLIDE 70

Classified Matchings under one sided preferences

Laminar classified popular matchings

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SLIDE 71

Classified Matchings under one sided preferences

Classified matchings: challenges

a1 a3 a2 p1 (2) p2(2) 1 1 M Deal with capacitated matchings.

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SLIDE 72

Classified Matchings under one sided preferences

Classified matchings: challenges

a1 a3 a2 p1 (2) p2(2) 1 1 M Deal with capacitated matchings.

Manlove and Sng use cloning. Paluch defined good paths.

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SLIDE 73

Classified Matchings under one sided preferences

Classified matchings: challenges

a1 a3 a2 p1 (2) p2(2) 1 1 M Deal with capacitated matchings.

Manlove and Sng use cloning. Paluch defined good paths.

Both techniques do not work for classifications.

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SLIDE 74

Classified Matchings under one sided preferences

Classified matchings: challenges

a1 a3 a2 p1 (2) p2(2) 1 1 M Deal with capacitated matchings.

Manlove and Sng use cloning. Paluch defined good paths.

Both techniques do not work for classifications.

Maximum matching M not feasible.

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SLIDE 75

Classified Matchings under one sided preferences

Classified matchings: challenges

a1 a3 a2 p1 (2) p2(2) 1 1 M Deal with capacitated matchings.

Manlove and Sng use cloning. Paluch defined good paths.

Both techniques do not work for classifications.

Maximum matching M not feasible.

Use max-flow and min-cut properties!

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SLIDE 76

Classified Matchings under one sided preferences

Properties of max-flow

Let H be any flow network and f be a max-flow in H. S = {v | is reachable from s in Hf } T = {v | v can reach t in Hf } U = {v | v / ∈ T ∪ S}

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SLIDE 77

Classified Matchings under one sided preferences

Properties of max-flow

Let H be any flow network and f be a max-flow in H. S = {v | is reachable from s in Hf } T = {v | v can reach t in Hf } U = {v | v / ∈ T ∪ S} (S, T ∪ U) is a min-s-t-cut in H

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SLIDE 78

Classified Matchings under one sided preferences

Properties of max-flow

Let H be any flow network and f be a max-flow in H. S = {v | is reachable from s in Hf } T = {v | v can reach t in Hf } U = {v | v / ∈ T ∪ S} (S, T ∪ U) is a min-s-t-cut in H Known Facts: Forward edges (S, T ∪ U) : saturated in every max-flow. Reverse edges (T ∪ U, S) : zero flow in every max-flow.

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SLIDE 79

Classified Matchings under one sided preferences

Properties of max-flow

s c d e t a b f g H S = {v | is reachable from s in Hf } T = {v | v can reach t in Hf } U = {v | v / ∈ T ∪ S}

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SLIDE 80

Classified Matchings under one sided preferences

Properties of max-flow

s c d e t a b f g Hf S = {v | is reachable from s in Hf } T = {v | v can reach t in Hf } U = {v | v / ∈ T ∪ S}

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SLIDE 81

Classified Matchings under one sided preferences

Properties of max-flow

The sets S, T and U are invariant of the max-flow f .

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SLIDE 82

Classified Matchings under one sided preferences

Properties of max-flow

The sets S, T and U are invariant of the max-flow f . (i) x ∈ Sf ⇐ ⇒ x ∈ Sf ′

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SLIDE 83

Classified Matchings under one sided preferences

Properties of max-flow

The sets S, T and U are invariant of the max-flow f . (i) x ∈ Sf ⇐ ⇒ x ∈ Sf ′ Proof: x ∈ Sf and x ∈ Tf ′ ∪ Uf ′

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SLIDE 84

Classified Matchings under one sided preferences

Properties of max-flow

The sets S, T and U are invariant of the max-flow f . (i) x ∈ Sf ⇐ ⇒ x ∈ Sf ′ Proof: x ∈ Sf and x ∈ Tf ′ ∪ Uf ′ x be the nearest such node from s in Hf .

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SLIDE 85

Classified Matchings under one sided preferences

Properties of max-flow

The sets S, T and U are invariant of the max-flow f . (i) x ∈ Sf ⇐ ⇒ x ∈ Sf ′ Proof: x ∈ Sf and x ∈ Tf ′ ∪ Uf ′ x be the nearest such node from s in Hf .

s y x Sf Tf ∪ Uf

Hf

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SLIDE 86

Classified Matchings under one sided preferences

Properties of max-flow

The sets S, T and U are invariant of the max-flow f . (i) x ∈ Sf ⇐ ⇒ x ∈ Sf ′ Proof: x ∈ Sf and x ∈ Tf ′ ∪ Uf ′ x be the nearest such node from s in Hf .

s y x Sf Tf ∪ Uf

Hf

s y x Sf ′ Tf ′ ∪ Uf ′

Hf ′ (y, x) ∈ H (x, y) ∈ H

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SLIDE 87

Classified Matchings under one sided preferences

Properties of max-flow

The sets S, T and U are invariant of the max-flow f . (i) x ∈ Sf ⇐ ⇒ x ∈ Sf ′ Proof: x ∈ Sf and x ∈ Tf ′ ∪ Uf ′ x be the nearest such node from s in Hf .

s y x Sf Tf ∪ Uf

Hf

s y x Sf ′ Tf ′ ∪ Uf ′

Hf ′ (y, x) ∈ H = ⇒ f (y, x) = f ′(y, x) = c(y, x). ⇒⇐ (x, y) ∈ H

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SLIDE 88

Classified Matchings under one sided preferences

Properties of max-flow

The sets S, T and U are invariant of the max-flow f . (i) x ∈ Sf ⇐ ⇒ x ∈ Sf ′ Proof: x ∈ Sf and x ∈ Tf ′ ∪ Uf ′ x be the nearest such node from s in Hf .

s y x Sf Tf ∪ Uf

Hf

s y x Sf ′ Tf ′ ∪ Uf ′

Hf ′ (y, x) ∈ H = ⇒ f (y, x) = f ′(y, x) = c(y, x). ⇒⇐ (x, y) ∈ H = ⇒ f (x, y) = f ′(x, y) = 0. ⇒⇐

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SLIDE 89

Classified Matchings under one sided preferences

Classified popular matchings: characterization

A matching M is popular if no matching beats it.

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SLIDE 90

Classified Matchings under one sided preferences

Classified popular matchings: characterization

A matching M is popular if no matching beats it. A matching M is popular if and only if: M is a maximum feasible matching on the rank-1 edges. Every a ∈ A is matched to either its f (a) or s(a).

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SLIDE 91

Classified Matchings under one sided preferences

Classified popular matchings: characterization

A matching M is popular if no matching beats it. A matching M is popular if and only if: M is a maximum feasible matching on the rank-1 edges. Every a ∈ A is matched to either its f (a) or s(a). f (a) - set of all rank-1 posts of a. s(a) - defined using flow network on rank-1 edges.

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SLIDE 92

Classified Matchings under one sided preferences

Classified popular matchings: characterization

A matching M is popular if no matching beats it. A matching M is popular if and only if: M is a maximum feasible matching on the rank-1 edges. Every a ∈ A is matched to either its f (a) or s(a). f (a) - set of all rank-1 posts of a. s(a) - defined using flow network on rank-1 edges. If such a matching does not exist, no popular matching exists.

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SLIDE 93

Classified Matchings under one sided preferences

Computing classified popular matchings

Overall idea: Reduction to two maximum flow computations.

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SLIDE 94

Classified Matchings under one sided preferences

Computing classified popular matchings

Overall idea: Reduction to two maximum flow computations.

1 Construct H1: every a adds (a, f (a)) edges.

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SLIDE 95

Classified Matchings under one sided preferences

Computing classified popular matchings

Overall idea: Reduction to two maximum flow computations.

1 Construct H1: every a adds (a, f (a)) edges. 2 Compute a maximum flow f1 in H1. 3 Delete (T ∪ U, S) edges in H1(f1).

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SLIDE 96

Classified Matchings under one sided preferences

Computing classified popular matchings

Overall idea: Reduction to two maximum flow computations.

1 Construct H1: every a adds (a, f (a)) edges. 2 Compute a maximum flow f1 in H1. 3 Delete (T ∪ U, S) edges in H1(f1).

Ensures that augmentation preserves max. card. on rank-1.

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SLIDE 97

Classified Matchings under one sided preferences

Computing classified popular matchings

Overall idea: Reduction to two maximum flow computations.

1 Construct H1: every a adds (a, f (a)) edges. 2 Compute a maximum flow f1 in H1. 3 Delete (T ∪ U, S) edges in H1(f1).

Ensures that augmentation preserves max. card. on rank-1.

4 For every a ∈ S, add (a, s(a)) edge.

s(a) is the most preferred post p ∈ T in H1(f1).

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SLIDE 98

Classified Matchings under one sided preferences

Computing classified popular matchings

Overall idea: Reduction to two maximum flow computations.

1 Construct H1: every a adds (a, f (a)) edges. 2 Compute a maximum flow f1 in H1. 3 Delete (T ∪ U, S) edges in H1(f1).

Ensures that augmentation preserves max. card. on rank-1.

4 For every a ∈ S, add (a, s(a)) edge.

s(a) is the most preferred post p ∈ T in H1(f1).

5 Augment f1 to obtain f2. Let M be matching corr. to f2.

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SLIDE 99

Classified Matchings under one sided preferences

Computing classified popular matchings

Overall idea: Reduction to two maximum flow computations.

1 Construct H1: every a adds (a, f (a)) edges. 2 Compute a maximum flow f1 in H1. 3 Delete (T ∪ U, S) edges in H1(f1).

Ensures that augmentation preserves max. card. on rank-1.

4 For every a ∈ S, add (a, s(a)) edge.

s(a) is the most preferred post p ∈ T in H1(f1).

5 Augment f1 to obtain f2. Let M be matching corr. to f2. 6 If M matches all applicants, declare popular,

else no popular matching.

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SLIDE 100

Classified Matchings under one sided preferences

Computing classified popular matchings

Overall idea: Reduction to two maximum flow computations.

1 Construct H1: every a adds (a, f (a)) edges. 2 Compute a maximum flow f1 in H1. 3 Delete (T ∪ U, S) edges in H1(f1).

Ensures that augmentation preserves max. card. on rank-1.

4 For every a ∈ S, add (a, s(a)) edge.

s(a) is the most preferred post p ∈ T in H1(f1).

5 Augment f1 to obtain f2. Let M be matching corr. to f2. 6 If M matches all applicants, declare popular,

else no popular matching. An O(|A| · |E|) time algorithm.

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SLIDE 101

Classified Matchings under one sided preferences

Classified popular matchings – correctness

A matching M is popular if and only if: M is a maximum feasible matching on the rank-1 edges. Every a ∈ A is matched to either its f (a) or s(a). If such a matching does not exist, no popular matching exists.

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SLIDE 102

Classified Matchings under one sided preferences

Classified popular matchings – correctness

A matching M is popular if and only if: M is a maximum feasible matching on the rank-1 edges. Every a ∈ A is matched to either its f (a) or s(a). If such a matching does not exist, no popular matching exists. Proof technique:

1 Two promotions at the cost of one demotion. 2 Cut edges were deleted =

⇒ augmentation preserves rank-1 edges.

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SLIDE 103

Classified Matchings under one sided preferences

To summarize ..

A flow based framework for popular matchings with classifications.

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SLIDE 104

Classified Matchings under one sided preferences

To summarize ..

A flow based framework for popular matchings with classifications. Same framework applies for rank-maximal matchings.

Classifications are allowed on both sides.

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SLIDE 105

Classified Matchings under one sided preferences

To summarize ..

A flow based framework for popular matchings with classifications. Same framework applies for rank-maximal matchings.

Classifications are allowed on both sides.

Key step: Use max-flow, min-cut properties to identify “unnecessary” edges.

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SLIDE 106

Classified Matchings under one sided preferences

To summarize ..

A flow based framework for popular matchings with classifications. Same framework applies for rank-maximal matchings.

Classifications are allowed on both sides.

Key step: Use max-flow, min-cut properties to identify “unnecessary” edges. Laminarity is needed, else problem is NP-Hard.

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SLIDE 107

Classified Matchings under one sided preferences

To summarize ..

A flow based framework for popular matchings with classifications. Same framework applies for rank-maximal matchings.

Classifications are allowed on both sides.

Key step: Use max-flow, min-cut properties to identify “unnecessary” edges. Laminarity is needed, else problem is NP-Hard. Extensions: Popular matchings in the many to many settings. Lower quotas for classes.

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SLIDE 108

Classified Matchings under one sided preferences

To summarize ..

A flow based framework for popular matchings with classifications. Same framework applies for rank-maximal matchings.

Classifications are allowed on both sides.

Key step: Use max-flow, min-cut properties to identify “unnecessary” edges. Laminarity is needed, else problem is NP-Hard. Extensions: Popular matchings in the many to many settings. Lower quotas for classes. Thank you!

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