Stability, Popularity, and Lower Quotas Meghana Nasre IIT Madras - - PowerPoint PPT Presentation

Stability, Popularity, and Lower Quotas

Stability, Popularity, and Lower Quotas

Meghana Nasre IIT Madras CAALM 2019 Chennai Mathematical Institute

Jan 22, 2019

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Stability, Popularity, and Lower Quotas

Problem Setup

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Stability, Popularity, and Lower Quotas

Problem Setup

A set of men A (applicants / students /medical interns)

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Stability, Popularity, and Lower Quotas

Problem Setup

A set of men A (applicants / students /medical interns) A set of women B ( jobs / courses / hospitals )

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Stability, Popularity, and Lower Quotas

Problem Setup

A set of men A (applicants / students /medical interns) A set of women B ( jobs / courses / hospitals ) Each participant has a preference ordering.

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Stability, Popularity, and Lower Quotas

Problem Setup

A set of men A (applicants / students /medical interns) A set of women B ( jobs / courses / hospitals ) Each participant has a preference ordering. a1: b1 b2 a2: b1 b2 b1: a1 a2 b2: a1 a2 Here preferences are strict and complete.

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Stability, Popularity, and Lower Quotas

Problem Setup

A set of men A (applicants / students /medical interns) A set of women B ( jobs / courses / hospitals ) Each participant has a preference ordering. a1: b1 b2 a2: b1 b2 b1: a1 a2 b2: a1 a2 Here preferences are strict and complete. Goal: Assign men to women optimally.

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Stability, Popularity, and Lower Quotas

Problem Setup

A set of men A (applicants / students /medical interns) A set of women B ( jobs / courses / hospitals ) Each participant has a preference ordering. a1: b1 b2 a2: b1 b2 b1: a1 a2 b2: a1 a2 Here preferences are strict and complete. A possible assignment M = {(a1, b2), (a2, b1)}.

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Stability, Popularity, and Lower Quotas

Stability – a notion of optimality

a1: b1 b2 a2: b1 b2 b1: a1 a2 b2: a1 a2 A pair (a, b) ∈ E \ M blocks M if Both a and b prefer each other to their current partner in M.

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Stability, Popularity, and Lower Quotas

Stability – a notion of optimality

a1: b1 b2 a2: b1 b2 b1: a1 a2 b2: a1 a2 A pair (a, b) ∈ E \ M blocks M if Both a and b prefer each other to their current partner in M. (a1, b1): both a1 and b1 wish to deviate – blocking pair.

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Stability, Popularity, and Lower Quotas

Stability – a notion of optimality

a1: b1 b2 a2: b1 b2 b1: a1 a2 b2: a1 a2 A pair (a, b) ∈ E \ M blocks M if Both a and b prefer each other to their current partner in M. (a1, b1): both a1 and b1 wish to deviate – blocking pair. A matching is stable if no pair wishes to deviate.

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Stability, Popularity, and Lower Quotas

Stability – a notion of optimality

a1: b1 b2 a2: b1 b2 b1: a1 a2 b2: a1 a2 A pair (a, b) ∈ E \ M blocks M if Both a and b prefer each other to their current partner in M. (a1, b1): both a1 and b1 wish to deviate – blocking pair. A matching is stable if no pair wishes to deviate. M′ = {(a1, b1), (a2, b2)} is a stable.

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Stability, Popularity, and Lower Quotas

Stability – a notion of optimality

a1: b1 b2 a2: b1 b2 b1: a1 a2 b2: a1 a2 A pair (a, b) ∈ E \ M blocks M if Both a and b prefer each other to their current partner in M. (a1, b1): both a1 and b1 wish to deviate – blocking pair. A matching is stable if no pair wishes to deviate. M′ = {(a1, b1), (a2, b2)} is a stable. Known Facts: Every instance admits a stable matching. Stable matching can be computed in linear time. All stable matchings are perfect.

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Stability, Popularity, and Lower Quotas

Today’s talk: Three Variants of the SM problem

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Stability, Popularity, and Lower Quotas

Variation #1: Incomplete Lists

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Stability, Popularity, and Lower Quotas

Variation #1: Incomplete Lists

Preferences are strict and can be incomplete. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1

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Stability, Popularity, and Lower Quotas

Variation #1: Incomplete Lists

Preferences are strict and can be incomplete. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Does a stable matching exist? Yes!

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Stability, Popularity, and Lower Quotas

Variation #1: Incomplete Lists

Preferences are strict and can be incomplete. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Does a stable matching exist? Yes! M = {(a1, b1)}.

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Stability, Popularity, and Lower Quotas

Variation #1: Incomplete Lists

Preferences are strict and can be incomplete. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Does a stable matching exist? Yes! M = {(a1, b1)}. Known Facts:

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Stability, Popularity, and Lower Quotas

Variation #1: Incomplete Lists

Preferences are strict and can be incomplete. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Does a stable matching exist? Yes! M = {(a1, b1)}. Known Facts: Every instance admits a stable matching; can be computed in linear time.

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Stability, Popularity, and Lower Quotas

Variation #1: Incomplete Lists

Preferences are strict and can be incomplete. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Does a stable matching exist? Yes! M = {(a1, b1)}. Known Facts: Every instance admits a stable matching; can be computed in linear time. All stable matchings are perfect

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Stability, Popularity, and Lower Quotas

Variation #1: Incomplete Lists

Preferences are strict and can be incomplete. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Does a stable matching exist? Yes! M = {(a1, b1)}. Known Facts: Every instance admits a stable matching; can be computed in linear time. All stable matchings are perfect of the same size.

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Stability, Popularity, and Lower Quotas

Variation #1: Incomplete Lists

Preferences are strict and can be incomplete. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Does a stable matching exist? Yes! M = {(a1, b1)}. Known Facts: Every instance admits a stable matching; can be computed in linear time. All stable matchings are perfect of the same size. Stable matching can be half the size of max. matching.

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Stability, Popularity, and Lower Quotas

Variation #1: Incomplete Lists

Preferences are strict and can be incomplete. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Does a stable matching exist? Yes! M = {(a1, b1)}. Known Facts: Every instance admits a stable matching; can be computed in linear time. All stable matchings are perfect of the same size. Stable matching can be half the size of max. matching. Question: Are there larger optimal matchings?

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Preferences can contain ties and can be incomplete. a1: b1 b2 a2: b1 b1: (a1 a2) b2: a1

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Preferences can contain ties and can be incomplete. a1: b1 b2 a2: b1 b1: (a1 a2) b2: a1 Redefine blocking pair.

A pair (a, b) ∈ E \ M blocks M if Both a and b strictly prefer each other to their current partner in M.

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Preferences can contain ties and can be incomplete. a1: b1 b2 a2: b1 b1: (a1 a2) b2: a1 Redefine blocking pair.

A pair (a, b) ∈ E \ M blocks M if Both a and b strictly prefer each other to their current partner in M.

Does a stable matching exist? Yes!

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Preferences can contain ties and can be incomplete. a1: b1 b2 a2: b1 b1: (a1 a2) b2: a1 Redefine blocking pair.

A pair (a, b) ∈ E \ M blocks M if Both a and b strictly prefer each other to their current partner in M.

Does a stable matching exist? Yes! M1 = {(a1, b1)} M2 = {(a1, b2), (a2, b1)}

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Preferences can contain ties and can be incomplete. a1: b1 b2 a2: b1 b1: (a1 a2) b2: a1 Redefine blocking pair.

A pair (a, b) ∈ E \ M blocks M if Both a and b strictly prefer each other to their current partner in M.

Does a stable matching exist? Yes! M1 = {(a1, b1)} M2 = {(a1, b2), (a2, b1)} Known Facts:

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Preferences can contain ties and can be incomplete. a1: b1 b2 a2: b1 b1: (a1 a2) b2: a1 Redefine blocking pair.

A pair (a, b) ∈ E \ M blocks M if Both a and b strictly prefer each other to their current partner in M.

Does a stable matching exist? Yes! M1 = {(a1, b1)} M2 = {(a1, b2), (a2, b1)} Known Facts: Every instance admits a stable matching; can be computed in linear time.

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Preferences can contain ties and can be incomplete. a1: b1 b2 a2: b1 b1: (a1 a2) b2: a1 Redefine blocking pair.

A pair (a, b) ∈ E \ M blocks M if Both a and b strictly prefer each other to their current partner in M.

Does a stable matching exist? Yes! M1 = {(a1, b1)} M2 = {(a1, b2), (a2, b1)} Known Facts: Every instance admits a stable matching; can be computed in linear time. All stable matchings are of the same size need not be of same size.

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Preferences can contain ties and can be incomplete. a1: b1 b2 a2: b1 b1: (a1 a2) b2: a1 Redefine blocking pair.

A pair (a, b) ∈ E \ M blocks M if Both a and b strictly prefer each other to their current partner in M.

Does a stable matching exist? Yes! M1 = {(a1, b1)} M2 = {(a1, b2), (a2, b1)} Known Facts: Every instance admits a stable matching; can be computed in linear time. All stable matchings are of the same size need not be of same size. A stable matching can be half the size of another stable matching.

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Preferences can contain ties and can be incomplete. a1: b1 b2 a2: b1 b1: (a1 a2) b2: a1 Redefine blocking pair.

A pair (a, b) ∈ E \ M blocks M if Both a and b strictly prefer each other to their current partner in M.

Does a stable matching exist? Yes! M1 = {(a1, b1)} M2 = {(a1, b2), (a2, b1)} Known Facts: Every instance admits a stable matching; can be computed in linear time. All stable matchings are of the same size need not be of same size. A stable matching can be half the size of another stable matching. Question: How to compute largest size stable matching?

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Stability, Popularity, and Lower Quotas

Variation #3: Lower Quotas

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Stability, Popularity, and Lower Quotas

Variation #3: Lower Quotas

Preferences are strict and can be incomplete. Some vertices must be matched – lower quota vertices.

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Stability, Popularity, and Lower Quotas

Variation #3: Lower Quotas

Preferences are strict and can be incomplete. Some vertices must be matched – lower quota vertices. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1

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Stability, Popularity, and Lower Quotas

Variation #3: Lower Quotas

Preferences are strict and can be incomplete. Some vertices must be matched – lower quota vertices. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Does a stable and feasible matching exist? Not necessarily.

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Stability, Popularity, and Lower Quotas

Variation #3: Lower Quotas

Preferences are strict and can be incomplete. Some vertices must be matched – lower quota vertices. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Does a stable and feasible matching exist? Not necessarily.

M1 = {(a1, b1)} Stable but not feasible. M1 = {(a2, b1), (a1, b2)} Feasible but not stable.

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Stability, Popularity, and Lower Quotas

Variation #3: Lower Quotas

Preferences are strict and can be incomplete. Some vertices must be matched – lower quota vertices. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Does a stable and feasible matching exist? Not necessarily.

M1 = {(a1, b1)} Stable but not feasible. M1 = {(a2, b1), (a1, b2)} Feasible but not stable.

Known Fact:

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Stability, Popularity, and Lower Quotas

Variation #3: Lower Quotas

Preferences are strict and can be incomplete. Some vertices must be matched – lower quota vertices. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Does a stable and feasible matching exist? Not necessarily.

M1 = {(a1, b1)} Stable but not feasible. M1 = {(a2, b1), (a1, b2)} Feasible but not stable.

Known Fact: In linear time we can check if an instance admits a feasible and stable matching.

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Stability, Popularity, and Lower Quotas

Variation #3: Lower Quotas

Preferences are strict and can be incomplete. Some vertices must be matched – lower quota vertices. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Does a stable and feasible matching exist? Not necessarily.

M1 = {(a1, b1)} Stable but not feasible. M1 = {(a2, b1), (a1, b2)} Feasible but not stable.

Known Fact: In linear time we can check if an instance admits a feasible and stable matching. Question: How to compute optimal feasible matching?

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Classical Model: Strict and Complete lists

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Computing a stable matching

Gale and Shapley 1962

a1: b1 b2 a2: b1 b2 b1: a1 a2 b2: a1 a2

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Stability, Popularity, and Lower Quotas

Computing a stable matching

Gale and Shapley 1962

a1: b1 b2 a2: b1 b2 b1: a1 a2 b2: a1 a2 Gale and Shapley Algo. Men propose. Women accept / reject.

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Stability, Popularity, and Lower Quotas

Computing a stable matching

Gale and Shapley 1962

a1: b1 b2 a2: b1 b2 b1: a1 a2 b2: a1 a2 Gale and Shapley Algo. Men propose. Women accept / reject. a1 → b1

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Stability, Popularity, and Lower Quotas

Computing a stable matching

Gale and Shapley 1962

a1: b1 b2 a2: b1 b2 b1: a1 a2 b2: a1 a2 Gale and Shapley Algo. Men propose. Women accept / reject. a1 → b1 accept.

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Stability, Popularity, and Lower Quotas

Computing a stable matching

Gale and Shapley 1962

a1: b1 b2 a2: b1 b2 b1: a1 a2 b2: a1 a2 Gale and Shapley Algo. Men propose. Women accept / reject. a1 → b1 accept. a2 → b1

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Stability, Popularity, and Lower Quotas

Computing a stable matching

Gale and Shapley 1962

a1: b1 b2 a2: b1 b2 b1: a1 a2 b2: a1 a2 Gale and Shapley Algo. Men propose. Women accept / reject. a1 → b1 accept. a2 → b1 reject.

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Stability, Popularity, and Lower Quotas

Computing a stable matching

Gale and Shapley 1962

a1: b1 b2 a2: b1 b2 b1: a1 a2 b2: a1 a2 Gale and Shapley Algo. Men propose. Women accept / reject. a1 → b1 accept. a2 → b1 reject. a2 → b2 accept.

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Stability, Popularity, and Lower Quotas

Computing a stable matching

Gale and Shapley 1962

a1: b1 b2 a2: b1 b2 b1: a1 a2 b2: a1 a2 Gale and Shapley Algo. Men propose. Women accept / reject. a1 → b1 accept. a2 → b1 reject. a2 → b2 accept.

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Stability, Popularity, and Lower Quotas

Computing a stable matching

Gale and Shapley 1962

a1: b1 b2 a2: b1 b2 b1: a1 a2 b2: a1 a2 Gale and Shapley Algo. Men propose. Women accept / reject. a1 → b1 accept. a2 → b1 reject. a2 → b2 accept. Ms = {(a1, b1), (a2, b2)}. Order of proposals does not matter. The side which proposes does matter.

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Stability, Popularity, and Lower Quotas

Models : Recap

Model Details Goal Classical setting strict and complete list Compute a stable match- ing

• Variation #1

strict and in- complete list Compute a larger optimal matching Variation #2 strict and tied list Compute a largest stable matching Variation #3 strict and in- complete list; lower quotas Compute a feasible

• pti-

mal matching

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Assume ties only on B side. a1: b1 b2 a2: b1 b1: (a1 a2) b2: a1

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Assume ties only on B side. a1: b1 b2 a2: b1 b1: (a1 a2) b2: a1 Recall: Multiple stable matchings of different sizes.

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Assume ties only on B side. a1: b1 b2 a2: b1 b1: (a1 a2) b2: a1 Recall: Multiple stable matchings of different sizes. M1 = {(a1, b1)} M2 = {(a1, b2), (a2, b1)} Compute largest size stable matching

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Assume ties only on B side. a1: b1 b2 a2: b1 b1: (a1 a2) b2: a1 Recall: Multiple stable matchings of different sizes. M1 = {(a1, b1)} M2 = {(a1, b2), (a2, b1)} Compute largest size stable matching NP-hard even for restricted setting.

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Assume ties only on B side. a1: b1 b2 a2: b1 b1: (a1 a2) b2: a1 Recall: Multiple stable matchings of different sizes. M1 = {(a1, b1)} M2 = {(a1, b2), (a2, b1)} Compute largest size stable matching NP-hard even for restricted setting. Naive method: Break ties arbitrarily. Run GS algo.

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Assume ties only on B side.

0.5 Irving and Manlove (2007) Kiraly (2011) Halldorsson et al. (2003) Iwama et al. (1999) Folklore 0.6 0.666 0.9 1

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Kir´ aly 2011

Assume ties only on B side. Kir´ aly’s Algorithm Break ties arbitrarily. Execute GS algo. Unmatched A’s propose again with increased priority.

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Kir´ aly 2011

Assume ties only on B side. Kir´ aly’s Algorithm Break ties arbitrarily. Execute GS algo. Unmatched A’s propose again with increased priority.

b uses increased priority for breaking ties.

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Kir´ aly 2011

Assume ties only on B side. Kir´ aly’s Algorithm Break ties arbitrarily. Execute GS algo. Unmatched A’s propose again with increased priority.

b uses increased priority for breaking ties. Stability is not violated.

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Kir´ aly 2011

a1: b1 b2 a2: b1 b1: (a1 a2) b2: a1

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Kir´ aly 2011

a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Kir´ aly’s Algo. Break ties arbitrarily. Run GS algo. Unmatched As propose with increased priority.

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Kir´ aly 2011

a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Kir´ aly’s Algo. Break ties arbitrarily. Run GS algo. Unmatched As propose with increased priority. a1 → b1

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Kir´ aly 2011

a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Kir´ aly’s Algo. Break ties arbitrarily. Run GS algo. Unmatched As propose with increased priority. a1 → b1 accept.

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Kir´ aly 2011

a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Kir´ aly’s Algo. Break ties arbitrarily. Run GS algo. Unmatched As propose with increased priority. a1 → b1 accept. a2 → b1

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Kir´ aly 2011

a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Kir´ aly’s Algo. Break ties arbitrarily. Run GS algo. Unmatched As propose with increased priority. a1 → b1 accept. a2 → b1 reject.

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Kir´ aly 2011

a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Kir´ aly’s Algo. Break ties arbitrarily. Run GS algo. Unmatched As propose with increased priority. a1 → b1 accept. a2 → b1 reject. a∗

2 → b1 accept; recall ties originally.

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Kir´ aly 2011

a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Kir´ aly’s Algo. Break ties arbitrarily. Run GS algo. Unmatched As propose with increased priority. a1 → b1 accept. a2 → b1 reject. a∗

2 → b1 accept; recall ties originally.

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Kir´ aly 2011

a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Kir´ aly’s Algo. Break ties arbitrarily. Run GS algo. Unmatched As propose with increased priority. a1 → b1 accept. a2 → b1 reject. a∗

2 → b1 accept; recall ties originally.

a1 → b2 accept.

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Kir´ aly 2011

a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Kir´ aly’s Algo. Break ties arbitrarily. Run GS algo. Unmatched As propose with increased priority. a1 → b1 accept. a2 → b1 reject. a∗

2 → b1 accept; recall ties originally.

a1 → b2 accept. M = {(a1, b2), (a2, b1)}.

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Kir´ aly 2011

a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Kir´ aly’s Algo. Break ties arbitrarily. Run GS algo. Unmatched As propose with increased priority. a1 → b1 accept. a2 → b1 reject. a∗

2 → b1 accept; recall ties originally.

a1 → b2 accept. M = {(a1, b2), (a2, b1)}. Goal: Argue about the size of the matching.

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Kir´ aly 2011

a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Kir´ aly’s Algo. Break ties arbitrarily. Run GS algo. Unmatched As propose with increased priority. a1 → b1 accept. a2 → b1 reject. a∗

2 → b1 accept; recall ties originally.

a1 → b2 accept. M = {(a1, b2), (a2, b1)}. Goal: Argue about the size of the matching. Show no short aug. paths.

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Matchings and aug. paths: a detour

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Matchings and aug. paths: a detour

Is this the largest sized matching?

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Stability, Popularity, and Lower Quotas

Matchings and aug. paths: a detour

Is this the largest sized matching? a2, b1, a3, b5 – alternating path with both end points free.

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Stability, Popularity, and Lower Quotas

Matchings and aug. paths: a detour

Is this the largest sized matching? a2, b1, a3, b5 – alternating path with both end points free.

• Aug. paths: odd number of

edges (1, 3, 5, ..., 2k+1)

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Stability, Popularity, and Lower Quotas

Matchings and aug. paths: a detour

Is this the largest sized matching? a2, b1, a3, b5 – alternating path with both end points free.

• Aug. paths: odd number of

edges (1, 3, 5, ..., 2k+1) No one length aug. path → maximal

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Stability, Popularity, and Lower Quotas

Matchings and aug. paths: a detour

Is this the largest sized matching? a2, b1, a3, b5 – alternating path with both end points free.

• Aug. paths: odd number of

edges (1, 3, 5, ..., 2k+1) No one length aug. path → maximal No short aug. path, closer to

• max. matching.

Matching M′ without 1 and 3 length aug. paths.

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Stability, Popularity, and Lower Quotas

Matchings and aug. paths: a detour

Is this the largest sized matching? a2, b1, a3, b5 – alternating path with both end points free.

• Aug. paths: odd number of

edges (1, 3, 5, ..., 2k+1) No one length aug. path → maximal No short aug. path, closer to

• max. matching.

Matching M′ without 1 and 3 length aug. paths. |M′| ≥ 2

3|M∗|.

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Back to Kir´ aly’s algorithm

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Recall Kir´ aly’s algorithm

Break ties arbitrarily. Execute GS algo. Unmatched A’s propose again with increased priority.

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Stability, Popularity, and Lower Quotas

Recall Kir´ aly’s algorithm

Break ties arbitrarily. Execute GS algo. Unmatched A’s propose again with increased priority.

b uses increased priority for breaking ties.

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Stability, Popularity, and Lower Quotas

Recall Kir´ aly’s algorithm

Break ties arbitrarily. Execute GS algo. Unmatched A’s propose again with increased priority.

b uses increased priority for breaking ties. Stability is not violated.

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Stability, Popularity, and Lower Quotas

Recall Kir´ aly’s algorithm

Break ties arbitrarily. Execute GS algo. Unmatched A’s propose again with increased priority.

b uses increased priority for breaking ties. Stability is not violated.

Need to argue about the size of the output.

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Stability, Popularity, and Lower Quotas

Recall Kir´ aly’s algorithm

Break ties arbitrarily. Execute GS algo. Unmatched A’s propose again with increased priority.

b uses increased priority for breaking ties. Stability is not violated.

Need to argue about the size of the output. Show that there are no short (1 and 3 length) aug. paths.

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Stability, Popularity, and Lower Quotas

Recall Kir´ aly’s algorithm

Break ties arbitrarily. Execute GS algo. Unmatched A’s propose again with increased priority.

b uses increased priority for breaking ties. Stability is not violated.

Need to argue about the size of the output. Show that there are no short (1 and 3 length) aug. paths. Some observations:

19 / 31

Stability, Popularity, and Lower Quotas

Recall Kir´ aly’s algorithm

Break ties arbitrarily. Execute GS algo. Unmatched A’s propose again with increased priority.

b uses increased priority for breaking ties. Stability is not violated.

Need to argue about the size of the output. Show that there are no short (1 and 3 length) aug. paths. Some observations: If a woman b is unmatched at the end of algo., she never got a proposal.

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Stability, Popularity, and Lower Quotas

Recall Kir´ aly’s algorithm

Break ties arbitrarily. Execute GS algo. Unmatched A’s propose again with increased priority.

b uses increased priority for breaking ties. Stability is not violated.

Need to argue about the size of the output. Show that there are no short (1 and 3 length) aug. paths. Some observations: If a woman b is unmatched at the end of algo., she never got a proposal. If a man a is unmatched at the end of algo., he got increased priority.

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Stability, Popularity, and Lower Quotas

Output of Kir´ aly’s algorithm

Suppose there exists a 3 length aug. path w.r.t. Malgo.

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Stability, Popularity, and Lower Quotas

Output of Kir´ aly’s algorithm

Suppose there exists a 3 length aug. path w.r.t. Malgo. a2 never proposed to b2 (∵ b2 is unmatched after algo) → a2 did not get increased priority.

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Stability, Popularity, and Lower Quotas

Output of Kir´ aly’s algorithm

Suppose there exists a 3 length aug. path w.r.t. Malgo. a2 never proposed to b2 (∵ b2 is unmatched after algo) → a2 did not get increased priority. → a2 strictly prefers b1 over b2.

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Stability, Popularity, and Lower Quotas

Output of Kir´ aly’s algorithm

Suppose there exists a 3 length aug. path w.r.t. Malgo. a2 never proposed to b2 (∵ b2 is unmatched after algo) → a2 did not get increased priority. → a2 strictly prefers b1 over b2. a1 is unmatched at the end of algo. → a1 must have got high priority.

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Stability, Popularity, and Lower Quotas

Output of Kir´ aly’s algorithm

Suppose there exists a 3 length aug. path w.r.t. Malgo. a2 never proposed to b2 (∵ b2 is unmatched after algo) → a2 did not get increased priority. → a2 strictly prefers b1 over b2. a1 is unmatched at the end of algo. → a1 must have got high priority. → b1 strictly prefers a2 over a1.

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Stability, Popularity, and Lower Quotas

Output of Kir´ aly’s algorithm

Suppose there exists a 3 length aug. path w.r.t. Malgo. a2 never proposed to b2 (∵ b2 is unmatched after algo) → a2 did not get increased priority. → a2 strictly prefers b1 over b2. a1 is unmatched at the end of algo. → a1 must have got high priority. → b1 strictly prefers a2 over a1. (a2, b1) is a blocking pair w.r.t. M∗.

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Stability, Popularity, and Lower Quotas

Output of Kir´ aly’s algorithm

Suppose there exists a 3 length aug. path w.r.t. Malgo. a2 never proposed to b2 (∵ b2 is unmatched after algo) → a2 did not get increased priority. → a2 strictly prefers b1 over b2. a1 is unmatched at the end of algo. → a1 must have got high priority. → b1 strictly prefers a2 over a1. (a2, b1) is a blocking pair w.r.t. M∗. contradicts stability of M∗. There are no 3 length aug. paths w.r.t. Malgo. Thus, |Malgo| ≥ 2

3|M∗|.

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Stability, Popularity, and Lower Quotas

Variation #2: Incomplete Lists and Ties

Assume ties only on B side.

0.5 Irving and Manlove (2007) Kiraly (2011) Halldorsson et al. (2003) Iwama et al. (1999) Folklore 0.6 0.666 0.9 1

Main takeaways A simple extension of GS algo. Extension to capacitated case (hospital residents). Extension to case of ties on both sides.

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Stability, Popularity, and Lower Quotas

Models : Recap

Model Details Goal Classical setting strict and complete list Compute a stable match- ing

• Variation #1

strict and in- complete list Compute a larger optimal matching Variation #2 strict and tied list Compute a largest stable matching

• Variation #3

strict and in- complete list; lower quotas Compute a feasible

• pti-

mal matching

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Stability, Popularity, and Lower Quotas

Variation #1 & Variation #3

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Stability, Popularity, and Lower Quotas

Popularity – an alternative to stability

G¨ ardenfors 1975

Compare two matchings by votes of participants.

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Stability, Popularity, and Lower Quotas

Popularity – an alternative to stability

G¨ ardenfors 1975

Compare two matchings by votes of participants. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Ms = {(a1, b1)} M = {(a2, b1)}

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Stability, Popularity, and Lower Quotas

Popularity – an alternative to stability

G¨ ardenfors 1975

Compare two matchings by votes of participants. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Ms = {(a1, b1)} M = {(a2, b1)} Ms M a1

• a2
• b1
• 24 / 31

Stability, Popularity, and Lower Quotas

Popularity – an alternative to stability

G¨ ardenfors 1975

Compare two matchings by votes of participants. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Ms = {(a1, b1)} M = {(a2, b1)} Ms M a1

• a2
• b1
• Ms beats M w.r.t. popularity.

Popular Matching: One which cannot be beaten!

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Stability, Popularity, and Lower Quotas

Popularity – an alternative to stability

G¨ ardenfors 1975

Compare two matchings by votes of participants. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Ms = {(a1, b1)} M = {(a2, b1)} Ms M a1

• a2
• b1
• Ms beats M w.r.t. popularity.

Popular Matching: One which cannot be beaten! Q: Does a popular matching exist?

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Stability, Popularity, and Lower Quotas

Popularity – an alternative to stability

G¨ ardenfors 1975

Compare two matchings by votes of participants. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Ms = {(a1, b1)} M′ = {(a1, b2), (a2, b1)}

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Stability, Popularity, and Lower Quotas

Popularity – an alternative to stability

G¨ ardenfors 1975

Compare two matchings by votes of participants. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Ms = {(a1, b1)} M′ = {(a1, b2), (a2, b1)} Ms M a1

• a2
• b1
• b2
• 25 / 31

Stability, Popularity, and Lower Quotas

Popularity – an alternative to stability

G¨ ardenfors 1975

Compare two matchings by votes of participants. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Ms = {(a1, b1)} M′ = {(a1, b2), (a2, b1)} Ms M a1

• a2
• b1
• b2
• Neither one beats each other.

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Stability, Popularity, and Lower Quotas

Popularity – an alternative to stability

G¨ ardenfors 1975

Compare two matchings by votes of participants. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Ms = {(a1, b1)} M′ = {(a1, b2), (a2, b1)} Ms M a1

• a2
• b1
• b2
• Neither one beats each other.

Does not prove popularity.

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Stability, Popularity, and Lower Quotas

Popularity – an alternative to stability

G¨ ardenfors 1975

Compare two matchings by votes of participants. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Ms = {(a1, b1)} M′ = {(a1, b2), (a2, b1)} Ms M a1

• a2
• b1
• b2
• Neither one beats each other.

Does not prove popularity. Q: Does a popular matching exist?

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Stability, Popularity, and Lower Quotas

Popularity – an alternative to stability

G¨ ardenfors 1975

Compare two matchings by votes of participants. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Ms = {(a1, b1)} M′ = {(a1, b2), (a2, b1)} Ms M a1

• a2
• b1
• b2
• Neither one beats each other.

Does not prove popularity. Q: Does a popular matching exist? Yes! A stable matching is popular.

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Stability, Popularity, and Lower Quotas

Variation #1: Incomplete Lists

Preferences are strict and can be incomplete. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Does a stable matching exist? Yes! M = {(a1, b1)}. Question: Are there larger optimal matchings?

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Stability, Popularity, and Lower Quotas

Variation #1: Incomplete Lists

Preferences are strict and can be incomplete. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Does a stable matching exist? Yes! M = {(a1, b1)}. Question: Are there larger optimal matchings? Yes! M′ = {(a1, b2), (a2, b1)} is popular.

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Stability, Popularity, and Lower Quotas

Variation #1: Incomplete Lists

Kavitha 2012

a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Goal: Compute Largest sized Popular matching.

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Stability, Popularity, and Lower Quotas

Variation #1: Incomplete Lists

Kavitha 2012

a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Goal: Compute Largest sized Popular matching. Run GS algo. Unmatched A’s propose with increased priority.

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Stability, Popularity, and Lower Quotas

Variation #1: Incomplete Lists

Kavitha 2012

a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Goal: Compute Largest sized Popular matching. Run GS algo. Unmatched A’s propose with increased priority. Stability may be violated.

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Stability, Popularity, and Lower Quotas

Variation #1: Incomplete Lists

Kavitha 2012

a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Goal: Compute Largest sized Popular matching. Run GS algo. Unmatched A’s propose with increased priority. Stability may be violated. Guarantees on output :

Malgo is max. sized popular. |Malgo| ≥ |Ms| and |Malgo| ≥ 2

3 |M∗|.

Linear time algo.

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Stability, Popularity, and Lower Quotas

Variation #1: Incomplete Lists

Kavitha 2012

a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Goal: Compute Largest sized Popular matching. Run GS algo. Unmatched A’s propose with increased priority. Stability may be violated. Guarantees on output :

Malgo is max. sized popular. |Malgo| ≥ |Ms| and |Malgo| ≥ 2

3 |M∗|.

Linear time algo.

Goal: Compute a max. card. matching that is popular.

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Stability, Popularity, and Lower Quotas

Variation #1: Incomplete Lists

Kavitha 2012

a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Goal: Compute Largest sized Popular matching. Run GS algo. Unmatched A’s propose with increased priority. Stability may be violated. Guarantees on output :

Malgo is max. sized popular. |Malgo| ≥ |Ms| and |Malgo| ≥ 2

3 |M∗|.

Linear time algo.

Goal: Compute a max. card. matching that is popular. Run GS algo. Unmatched A’s propose with increased priority n times.

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Stability, Popularity, and Lower Quotas

Variation #1: Incomplete Lists

Kavitha 2012

a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Goal: Compute Largest sized Popular matching. Run GS algo. Unmatched A’s propose with increased priority. Stability may be violated. Guarantees on output :

Malgo is max. sized popular. |Malgo| ≥ |Ms| and |Malgo| ≥ 2

3 |M∗|.

Linear time algo.

Goal: Compute a max. card. matching that is popular. Run GS algo. Unmatched A’s propose with increased priority n times. Stability may be violated.

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Stability, Popularity, and Lower Quotas

Variation #1: Incomplete Lists

Kavitha 2012

a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Goal: Compute Largest sized Popular matching. Run GS algo. Unmatched A’s propose with increased priority. Stability may be violated. Guarantees on output :

Malgo is max. sized popular. |Malgo| ≥ |Ms| and |Malgo| ≥ 2

3 |M∗|.

Linear time algo.

Goal: Compute a max. card. matching that is popular. Run GS algo. Unmatched A’s propose with increased priority n times. Stability may be violated. Guarantees on output :

Malgo is max. cardinality matching. Malgo is popular amongst max. card. matchings. Running time: O(nm).

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Stability, Popularity, and Lower Quotas

Variation #3: Lower Quotas

Preferences are strict and can be incomplete. Some vertices must be matched – lower quota vertices. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Does a stable and feasible matching exist? Not necessarily. Question: How to compute optimal feasible matching?

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Stability, Popularity, and Lower Quotas

Variation #3: Lower Quotas

Preferences are strict and can be incomplete. Some vertices must be matched – lower quota vertices. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Does a stable and feasible matching exist? Not necessarily. Question: How to compute optimal feasible matching? Yes! M′ = {(a2, b1), (a1, b2)} Feasible, not stable, but popular.

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Stability, Popularity, and Lower Quotas

Variation #3: Lower Quotas

N., Nimbhorkar 2017

Preferences are strict and can be incomplete. Some vertices must be matched – lower quota vertices. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Goal: Compute a feasible matching that is popular.

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Stability, Popularity, and Lower Quotas

Variation #3: Lower Quotas

N., Nimbhorkar 2017

Preferences are strict and can be incomplete. Some vertices must be matched – lower quota vertices. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Goal: Compute a feasible matching that is popular. Run GS algo. Unmatched A’s propose with increased priority. Deficient A’s propose as long as they are deficient.

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Stability, Popularity, and Lower Quotas

Variation #3: Lower Quotas

N., Nimbhorkar 2017

Preferences are strict and can be incomplete. Some vertices must be matched – lower quota vertices. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Goal: Compute a feasible matching that is popular. Run GS algo. Unmatched A’s propose with increased priority. Deficient A’s propose as long as they are deficient. Stability will be violated.

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Stability, Popularity, and Lower Quotas

Variation #3: Lower Quotas

N., Nimbhorkar 2017

Preferences are strict and can be incomplete. Some vertices must be matched – lower quota vertices. a1: b1 b2 a2: b1 b1: a1 a2 b2: a1 Goal: Compute a feasible matching that is popular. Run GS algo. Unmatched A’s propose with increased priority. Deficient A’s propose as long as they are deficient. Stability will be violated. Guarantees on output :

Malgo is feasible. Malgo is max. sized popular. amongst feasible matchings. Running time: O(nm).

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Stability, Popularity, and Lower Quotas

Models : Summary

Model Details Goal Classical setting strict and complete list Compute a stable match- ing

• Variation #1

strict and in- complete list Compute a larger optimal matching

• Variation #2

strict and tied list Compute a largest stable matching

• Variation #3

strict and in- complete list; lower quotas Compute a feasible

• pti-

mal matching

• 30 / 31

Stability, Popularity, and Lower Quotas

To summarize..

A simple extension of GS algo. Each case requires different proof techniques and several non-trivial details. All algorithms can be written as a reduction to a suitable SM instance. Works in the presence of capacities (upper quotas).

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Stability, Popularity, and Lower Quotas

To summarize..

A simple extension of GS algo. Each case requires different proof techniques and several non-trivial details. All algorithms can be written as a reduction to a suitable SM instance. Works in the presence of capacities (upper quotas).

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Stability, Popularity, and Lower Quotas

To summarize..

A simple extension of GS algo. Each case requires different proof techniques and several non-trivial details. All algorithms can be written as a reduction to a suitable SM instance. Works in the presence of capacities (upper quotas). Thank You!

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