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

Stability, Popularity, and Lower Quotas Variation #2: Incomplete Lists and Ties Preferences can contain ties and can be incomplete. a 1 : b 1 : ( a 1 a 2 ) b 1 b 2 a 2 : b 2 : b 1 a 1 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! M 1 = { ( a 1 , b 1 ) } M 2 = { ( a 1 , b 2 ) , ( a 2 , b 1 ) } 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. 7 / 31

Stability, Popularity, and Lower Quotas Variation #2: Incomplete Lists and Ties Preferences can contain ties and can be incomplete. a 1 : b 1 : ( a 1 a 2 ) b 1 b 2 a 2 : b 2 : b 1 a 1 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! M 1 = { ( a 1 , b 1 ) } M 2 = { ( a 1 , b 2 ) , ( a 2 , b 1 ) } 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. 7 / 31

Stability, Popularity, and Lower Quotas Variation #2: Incomplete Lists and Ties Preferences can contain ties and can be incomplete. a 1 : b 1 : ( a 1 a 2 ) b 1 b 2 a 2 : b 2 : b 1 a 1 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! M 1 = { ( a 1 , b 1 ) } M 2 = { ( a 1 , b 2 ) , ( a 2 , b 1 ) } 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? 7 / 31

Stability, Popularity, and Lower Quotas Variation #3: Lower Quotas 8 / 31

Stability, Popularity, and Lower Quotas Variation #3: Lower Quotas Preferences are strict and can be incomplete. Some vertices must be matched – lower quota vertices. 8 / 31

Stability, Popularity, and Lower Quotas Variation #3: Lower Quotas Preferences are strict and can be incomplete. Some vertices must be matched – lower quota vertices. a 1 : b 1 : b 1 b 2 a 1 a 2 a 2 : b 2 : b 1 a 1 8 / 31

Stability, Popularity, and Lower Quotas Variation #3: Lower Quotas Preferences are strict and can be incomplete. Some vertices must be matched – lower quota vertices. a 1 : b 1 : b 1 b 2 a 1 a 2 a 2 : b 2 : b 1 a 1 Does a stable and feasible matching exist? Not necessarily. 8 / 31

Stability, Popularity, and Lower Quotas Variation #3: Lower Quotas Preferences are strict and can be incomplete. Some vertices must be matched – lower quota vertices. a 1 : b 1 : b 1 b 2 a 1 a 2 a 2 : b 2 : b 1 a 1 Does a stable and feasible matching exist? Not necessarily. M 1 = { ( a 1 , b 1 ) } Stable but not feasible. M 1 = { ( a 2 , b 1 ) , ( a 1 , b 2 ) } Feasible but not stable. 8 / 31

Stability, Popularity, and Lower Quotas Variation #3: Lower Quotas Preferences are strict and can be incomplete. Some vertices must be matched – lower quota vertices. a 1 : b 1 : b 1 b 2 a 1 a 2 a 2 : b 2 : b 1 a 1 Does a stable and feasible matching exist? Not necessarily. M 1 = { ( a 1 , b 1 ) } Stable but not feasible. M 1 = { ( a 2 , b 1 ) , ( a 1 , b 2 ) } Feasible but not stable. Known Fact: 8 / 31

Stability, Popularity, and Lower Quotas Variation #3: Lower Quotas Preferences are strict and can be incomplete. Some vertices must be matched – lower quota vertices. a 1 : b 1 : b 1 b 2 a 1 a 2 a 2 : b 2 : b 1 a 1 Does a stable and feasible matching exist? Not necessarily. M 1 = { ( a 1 , b 1 ) } Stable but not feasible. M 1 = { ( a 2 , b 1 ) , ( a 1 , b 2 ) } Feasible but not stable. Known Fact: In linear time we can check if an instance admits a feasible and stable matching. 8 / 31

Stability, Popularity, and Lower Quotas Variation #3: Lower Quotas Preferences are strict and can be incomplete. Some vertices must be matched – lower quota vertices. a 1 : b 1 : b 1 b 2 a 1 a 2 a 2 : b 2 : b 1 a 1 Does a stable and feasible matching exist? Not necessarily. M 1 = { ( a 1 , b 1 ) } Stable but not feasible. M 1 = { ( a 2 , b 1 ) , ( a 1 , b 2 ) } 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? 8 / 31

Stability, Popularity, and Lower Quotas Classical Model: Strict and Complete lists 9 / 31

Stability, Popularity, and Lower Quotas Computing a stable matching Gale and Shapley 1962 a 1 : b 1 b 2 b 1 : a 1 a 2 a 2 : b 1 b 2 b 2 : a 1 a 2 10 / 31

Stability, Popularity, and Lower Quotas Computing a stable matching Gale and Shapley 1962 a 1 : b 1 b 2 b 1 : a 1 a 2 a 2 : b 1 b 2 b 2 : a 1 a 2 Gale and Shapley Algo. Men propose. Women accept / reject. 10 / 31

Stability, Popularity, and Lower Quotas Computing a stable matching Gale and Shapley 1962 a 1 : b 1 b 2 b 1 : a 1 a 2 a 2 : b 1 b 2 b 2 : a 1 a 2 Gale and Shapley Algo. a 1 → b 1 Men propose. Women accept / reject. 10 / 31

Stability, Popularity, and Lower Quotas Computing a stable matching Gale and Shapley 1962 a 1 : b 1 b 2 b 1 : a 1 a 2 a 2 : b 1 b 2 b 2 : a 1 a 2 Gale and Shapley Algo. a 1 → b 1 accept. Men propose. Women accept / reject. 10 / 31

Stability, Popularity, and Lower Quotas Computing a stable matching Gale and Shapley 1962 a 1 : b 1 b 2 b 1 : a 1 a 2 a 2 : b 1 b 2 b 2 : a 1 a 2 Gale and Shapley Algo. a 1 → b 1 accept. Men propose. a 2 → b 1 Women accept / reject. 10 / 31

Stability, Popularity, and Lower Quotas Computing a stable matching Gale and Shapley 1962 a 1 : b 1 b 2 b 1 : a 1 a 2 a 2 : b 1 b 2 b 2 : a 1 a 2 Gale and Shapley Algo. a 1 → b 1 accept. Men propose. a 2 → b 1 reject. Women accept / reject. 10 / 31

Stability, Popularity, and Lower Quotas Computing a stable matching Gale and Shapley 1962 a 1 : b 1 b 2 b 1 : a 1 a 2 a 2 : b 1 b 2 b 2 : a 1 a 2 Gale and Shapley Algo. a 1 → b 1 accept. Men propose. a 2 → b 1 reject. Women accept / reject. a 2 → b 2 accept. 10 / 31

Stability, Popularity, and Lower Quotas Computing a stable matching Gale and Shapley 1962 a 1 : b 1 b 2 b 1 : a 1 a 2 a 2 : b 1 b 2 b 2 : a 1 a 2 Gale and Shapley Algo. a 1 → b 1 accept. Men propose. a 2 → b 1 reject. Women accept / reject. a 2 → b 2 accept. 10 / 31

Stability, Popularity, and Lower Quotas Computing a stable matching Gale and Shapley 1962 a 1 : b 1 b 2 b 1 : a 1 a 2 a 2 : b 1 b 2 b 2 : a 1 a 2 a 1 → b 1 accept. Gale and Shapley Algo. a 2 → b 1 reject. Men propose. a 2 → b 2 accept. Women accept / reject. M s = { ( a 1 , b 1 ) , ( a 2 , b 2 ) } . Order of proposals does not matter. The side which proposes does matter. 10 / 31

Stability, Popularity, and Lower Quotas Models : Recap Model Details Goal Classical strict and Compute a � setting complete list stable match- ing Variation #1 strict and in- Compute a complete list larger optimal matching Variation #2 strict and tied Compute a list largest stable matching Variation #3 strict and in- Compute a complete list; feasible opti- lower quotas mal matching 11 / 31

Stability, Popularity, and Lower Quotas Variation #2: Incomplete Lists and Ties 12 / 31

Stability, Popularity, and Lower Quotas Variation #2: Incomplete Lists and Ties Assume ties only on B side. a 1 : b 1 b 2 b 1 : ( a 1 a 2 ) a 2 : b 1 b 2 : a 1 13 / 31

Stability, Popularity, and Lower Quotas Variation #2: Incomplete Lists and Ties Assume ties only on B side. a 1 : b 1 b 2 b 1 : ( a 1 a 2 ) a 2 : b 1 b 2 : a 1 Recall: Multiple stable matchings of different sizes. 13 / 31

Stability, Popularity, and Lower Quotas Variation #2: Incomplete Lists and Ties Assume ties only on B side. a 1 : b 1 b 2 b 1 : ( a 1 a 2 ) a 2 : b 1 b 2 : a 1 Recall: Multiple stable matchings of different sizes. M 1 = { ( a 1 , b 1 ) } M 2 = { ( a 1 , b 2 ) , ( a 2 , b 1 ) } Compute largest size stable matching 13 / 31

Stability, Popularity, and Lower Quotas Variation #2: Incomplete Lists and Ties Assume ties only on B side. a 1 : b 1 b 2 b 1 : ( a 1 a 2 ) a 2 : b 1 b 2 : a 1 Recall: Multiple stable matchings of different sizes. M 1 = { ( a 1 , b 1 ) } M 2 = { ( a 1 , b 2 ) , ( a 2 , b 1 ) } Compute largest size stable matching NP-hard even for restricted setting. 13 / 31

Stability, Popularity, and Lower Quotas Variation #2: Incomplete Lists and Ties Assume ties only on B side. a 1 : b 1 b 2 b 1 : ( a 1 a 2 ) a 2 : b 1 b 2 : a 1 Recall: Multiple stable matchings of different sizes. M 1 = { ( a 1 , b 1 ) } M 2 = { ( a 1 , b 2 ) , ( a 2 , b 1 ) } Compute largest size stable matching NP-hard even for restricted setting. Naive method: Break ties arbitrarily. Run GS algo. 13 / 31

Stability, Popularity, and Lower Quotas Variation #2: Incomplete Lists and Ties Assume ties only on B side. Irving and Manlove (2007) Folklore 0.666 0.9 1 0 0.5 0.6 Iwama et al. Kiraly (1999) (2011) Halldorsson et al. (2003) 14 / 31

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. 15 / 31

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. 15 / 31

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. 15 / 31

Stability, Popularity, and Lower Quotas Variation #2: Incomplete Lists and Ties Kir´ aly 2011 a 1 : b 1 b 2 b 1 : ( a 1 a 2 ) a 2 : b 1 b 2 : a 1 16 / 31

Stability, Popularity, and Lower Quotas Variation #2: Incomplete Lists and Ties Kir´ aly 2011 a 1 : b 1 b 2 b 1 : a 1 a 2 a 2 : b 1 b 2 : a 1 Kir´ aly’s Algo. Break ties arbitrarily. Run GS algo. Unmatched A s propose with increased priority. 16 / 31

Stability, Popularity, and Lower Quotas Variation #2: Incomplete Lists and Ties Kir´ aly 2011 a 1 : b 1 b 2 b 1 : a 1 a 2 a 2 : b 1 b 2 : a 1 Kir´ aly’s Algo. a 1 → b 1 Break ties arbitrarily. Run GS algo. Unmatched A s propose with increased priority. 16 / 31

Stability, Popularity, and Lower Quotas Variation #2: Incomplete Lists and Ties Kir´ aly 2011 a 1 : b 1 b 2 b 1 : a 1 a 2 a 2 : b 1 b 2 : a 1 Kir´ aly’s Algo. a 1 → b 1 accept. Break ties arbitrarily. Run GS algo. Unmatched A s propose with increased priority. 16 / 31

Stability, Popularity, and Lower Quotas Variation #2: Incomplete Lists and Ties Kir´ aly 2011 a 1 : b 1 b 2 b 1 : a 1 a 2 a 2 : b 1 b 2 : a 1 Kir´ aly’s Algo. a 1 → b 1 accept. Break ties arbitrarily. a 2 → b 1 Run GS algo. Unmatched A s propose with increased priority. 16 / 31

Stability, Popularity, and Lower Quotas Variation #2: Incomplete Lists and Ties Kir´ aly 2011 a 1 : b 1 b 2 b 1 : a 1 a 2 a 2 : b 1 b 2 : a 1 Kir´ aly’s Algo. a 1 → b 1 accept. Break ties arbitrarily. a 2 → b 1 reject. Run GS algo. Unmatched A s propose with increased priority. 16 / 31

Stability, Popularity, and Lower Quotas Variation #2: Incomplete Lists and Ties Kir´ aly 2011 a 1 : b 1 b 2 b 1 : a 1 a 2 a 2 : b 1 b 2 : a 1 Kir´ aly’s Algo. a 1 → b 1 accept. Break ties arbitrarily. a 2 → b 1 reject. Run GS algo. a ∗ 2 → b 1 accept; recall ties originally. Unmatched A s propose with increased priority. 16 / 31

Stability, Popularity, and Lower Quotas Variation #2: Incomplete Lists and Ties Kir´ aly 2011 a 1 : b 1 b 2 b 1 : a 1 a 2 a 2 : b 1 b 2 : a 1 Kir´ aly’s Algo. a 1 → b 1 accept. Break ties arbitrarily. a 2 → b 1 reject. Run GS algo. a ∗ 2 → b 1 accept; recall ties originally. Unmatched A s propose with increased priority. 16 / 31

Stability, Popularity, and Lower Quotas Variation #2: Incomplete Lists and Ties Kir´ aly 2011 a 1 : b 1 b 2 b 1 : a 1 a 2 a 2 : b 1 b 2 : a 1 Kir´ aly’s Algo. a 1 → b 1 accept. Break ties arbitrarily. a 2 → b 1 reject. Run GS algo. a ∗ 2 → b 1 accept; recall ties originally. Unmatched A s propose with increased priority. a 1 → b 2 accept. 16 / 31

Stability, Popularity, and Lower Quotas Variation #2: Incomplete Lists and Ties Kir´ aly 2011 a 1 : b 1 b 2 b 1 : a 1 a 2 a 2 : b 1 b 2 : a 1 Kir´ aly’s Algo. a 1 → b 1 accept. Break ties arbitrarily. a 2 → b 1 reject. Run GS algo. a ∗ 2 → b 1 accept; recall ties originally. Unmatched A s propose with a 1 → b 2 accept. increased priority. M = { ( a 1 , b 2 ) , ( a 2 , b 1 ) } . 16 / 31

Stability, Popularity, and Lower Quotas Variation #2: Incomplete Lists and Ties Kir´ aly 2011 a 1 : b 1 b 2 b 1 : a 1 a 2 a 2 : b 1 b 2 : a 1 Kir´ aly’s Algo. a 1 → b 1 accept. Break ties arbitrarily. a 2 → b 1 reject. Run GS algo. a ∗ 2 → b 1 accept; recall ties originally. Unmatched A s propose with a 1 → b 2 accept. increased priority. M = { ( a 1 , b 2 ) , ( a 2 , b 1 ) } . Goal: Argue about the size of the matching. 16 / 31

Stability, Popularity, and Lower Quotas Variation #2: Incomplete Lists and Ties Kir´ aly 2011 a 1 : b 1 b 2 b 1 : a 1 a 2 a 2 : b 1 b 2 : a 1 Kir´ aly’s Algo. a 1 → b 1 accept. Break ties arbitrarily. a 2 → b 1 reject. Run GS algo. a ∗ 2 → b 1 accept; recall ties originally. Unmatched A s propose with a 1 → b 2 accept. increased priority. M = { ( a 1 , b 2 ) , ( a 2 , b 1 ) } . Goal: Argue about the size of the matching. Show no short aug. paths. 16 / 31

Stability, Popularity, and Lower Quotas Matchings and aug. paths: a detour 17 / 31

Stability, Popularity, and Lower Quotas Matchings and aug. paths: a detour Is this the largest sized matching? 17 / 31

Stability, Popularity, and Lower Quotas Matchings and aug. paths: a detour Is this the largest sized matching? a 2 , b 1 , a 3 , b 5 – alternating path with both end points free. 17 / 31

Stability, Popularity, and Lower Quotas Matchings and aug. paths: a detour Is this the largest sized matching? a 2 , b 1 , a 3 , b 5 – alternating path with both end points free. Aug. paths: odd number of edges (1, 3, 5, ..., 2k+1) 17 / 31

Stability, Popularity, and Lower Quotas Matchings and aug. paths: a detour Is this the largest sized matching? a 2 , b 1 , a 3 , b 5 – alternating path with both end points free. Aug. paths: odd number of edges (1, 3, 5, ..., 2k+1) No one length aug. path → maximal 17 / 31

Stability, Popularity, and Lower Quotas Matchings and aug. paths: a detour Is this the largest sized matching? a 2 , b 1 , a 3 , b 5 – 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. 17 / 31

Stability, Popularity, and Lower Quotas Matchings and aug. paths: a detour Is this the largest sized matching? a 2 , b 1 , a 3 , b 5 – 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 ∗ | . 17 / 31

Stability, Popularity, and Lower Quotas Back to Kir´ aly’s algorithm 18 / 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. 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. 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. 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. 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. 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: 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. 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. If a man a is unmatched at the end of algo., he got increased priority. 19 / 31

Stability, Popularity, and Lower Quotas Output of Kir´ aly’s algorithm Suppose there exists a 3 length aug. path w.r.t. M algo . 20 / 31

Stability, Popularity, and Lower Quotas Output of Kir´ aly’s algorithm Suppose there exists a 3 length aug. path w.r.t. M algo . a 2 never proposed to b 2 ( ∵ b 2 is unmatched after algo) → a 2 did not get increased priority. 20 / 31

Stability, Popularity, and Lower Quotas Output of Kir´ aly’s algorithm Suppose there exists a 3 length aug. path w.r.t. M algo . a 2 never proposed to b 2 ( ∵ b 2 is unmatched after algo) → a 2 did not get increased priority. → a 2 strictly prefers b 1 over b 2 . 20 / 31

Stability, Popularity, and Lower Quotas Output of Kir´ aly’s algorithm Suppose there exists a 3 length aug. path w.r.t. M algo . a 2 never proposed to b 2 ( ∵ b 2 is unmatched after algo) → a 2 did not get increased priority. → a 2 strictly prefers b 1 over b 2 . a 1 is unmatched at the end of algo. → a 1 must have got high priority. 20 / 31

Stability, Popularity, and Lower Quotas Output of Kir´ aly’s algorithm Suppose there exists a 3 length aug. path w.r.t. M algo . a 2 never proposed to b 2 ( ∵ b 2 is unmatched after algo) → a 2 did not get increased priority. → a 2 strictly prefers b 1 over b 2 . a 1 is unmatched at the end of algo. → a 1 must have got high priority. → b 1 strictly prefers a 2 over a 1 . 20 / 31

Stability, Popularity, and Lower Quotas Output of Kir´ aly’s algorithm Suppose there exists a 3 length aug. path w.r.t. M algo . a 2 never proposed to b 2 ( ∵ b 2 is unmatched after algo) → a 2 did not get increased priority. → a 2 strictly prefers b 1 over b 2 . a 1 is unmatched at the end of algo. → a 1 must have got high priority. → b 1 strictly prefers a 2 over a 1 . ( a 2 , b 1 ) is a blocking pair w.r.t. M ∗ . 20 / 31

Stability, Popularity, and Lower Quotas Output of Kir´ aly’s algorithm Suppose there exists a 3 length aug. path w.r.t. M algo . a 2 never proposed to b 2 ( ∵ b 2 is unmatched after algo) → a 2 did not get increased priority. → a 2 strictly prefers b 1 over b 2 . a 1 is unmatched at the end of algo. → a 1 must have got high priority. → b 1 strictly prefers a 2 over a 1 . ( a 2 , b 1 ) is a blocking pair w.r.t. M ∗ . contradicts stability of M ∗ . There are no 3 length aug. paths w.r.t. M algo . Thus, | M algo | ≥ 2 3 | M ∗ | . 20 / 31

Stability, Popularity, and Lower Quotas Variation #2: Incomplete Lists and Ties Assume ties only on B side. Irving and Manlove (2007) Folklore 0.666 0.9 1 0 0.5 0.6 Iwama et al. Kiraly (1999) (2011) Halldorsson et al. (2003) Main takeaways A simple extension of GS algo. Extension to capacitated case (hospital residents). Extension to case of ties on both sides. 21 / 31

Stability, Popularity, and Lower Quotas Models : Recap Model Details Goal Classical strict and Compute a � setting complete list stable match- ing Variation #1 strict and in- Compute a complete list larger optimal matching Variation #2 strict and tied Compute a � list largest stable matching Variation #3 strict and in- Compute a complete list; feasible opti- lower quotas mal matching 22 / 31