Socially stable matchings in the Hospitals/Residents problem

Socially stable matchings in the Hospitals/Residents problem a.kwanashie.1@research.gla.ac.uk Joint work with Georgios Askalidis1, Nicole Immorlica1, David Manlove2 and Emmanouil Pountourakis1 1Northwestern University 2University of Glasgow AugustineKwanashie2

Content • Introduction to matching problems • Motivation for the Hospitals/Residents problem under Social Stability (HRSS) and MAX HRSS • Approximating MAX HRSS • Future work

Matching Problems • Generally involve the assignment of agents from one set to those of another • Assigning medical graduates to hospital placements • Assigning final year students to projects • Assigning kidney donors to recipients • Agents in all (or some) of the sets involved may be required to list other agents they find acceptable in order of preference • A matchingis a set of such assignments that meets some criteria • may take a one-to-one or many-to-one form • Problem is to find a stablematching • one in which no two agents, who are not matched to each other, prefer each other to their current partners • they may be tempted to ignore their assigned partners

The Stable Marriage Problem (SM) Men Women 1: 2 4 1 3 1: 2 1 4 3 2: 3 1 4 2 2: 4 3 1 2 3: 2 3 1 4 3: 1 4 3 2 4: 4 1 3 2 4: 2 1 4 3 Stable Matching M = {(1,4), (2,3), (3,2), (4,1)} • Man 1preferswoman 2 to woman 4 and prefers woman 4 to woman 1 • Matching M is a set of man-woman pairs where everyone appears exactly once • Matching M is stable if no man-woman pair prefer each other to their assignment in M • Every instance admits a stable matching (Gale and Shapley ‘62)

The Hospitals/Residents Problem (HR) Residents Hospitals 1: 2 1 1: (2): 1 3 2 5 6 2: 1 2 2: (2): 2 6 1 4 5 3: 1 3 3: (2): 4 3 4: 2 3 5: 2 1 6: 1 2 Stable Matching M = {(1,2), (2,1), (3,1), (4,3), (6,2)} • AHR instance Iconsists of a set of residentsR, a set of hospitalsHand a set of acceptable pairsE • Matching M is a set of resident-hospital pairs where each resident appears at most once and each hospital does not exceed its capacity • All preferences are reciprocal • We say (r, h) is an acceptable pair if r and h appear on each others’ preference list

The Hospitals/Residents Problem (HR) Residents Hospitals 1: 2 1 1: (2): 1 3 2 5 6 2: 1 2 2: (2): 2 6 1 4 5 3: 1 3 3: (2): 4 3 4: 2 3 5: 2 1 6: 1 2 Stable Matching M = {(1,2), (2,1), (3,1), (4,3), (6,2)} • Matching M is stable if there is no resident-hospital pair (r, h) such that: • r is either unmatchedor prefers h to her partner in M and • h is either undersubscribedor prefers r to one of its assigned residents • We say (r, h)blocksM or forms a blocking pair with respect to M

The Hospitals/Residents Problem (HR) Residents Hospitals 1: 2 1 1: (2): 1 3 2 5 6 2: 1 2 2: (2): 2 6 1 4 5 3: 1 3 3: (2): 4 3 4: 2 3 5: 2 1 6: 1 2 Stable Matching M = {(1,2), (2,1), (3,1), (4,3), (6,2)} |M| = 5 • Stable matchings in HR instances need not be complete • but all stable matchings in a given instance have the same size • all stable matchings in a given instance match the same set of residents • all undersubscribed hospitals in a given stable matching will remain matched to the same set of residents in all stable matchings (Roth ’84, Gale and Sotomayor ‘85, Roth ‘86)

Hospitals/Residents Problem under Social Stability (HRSS) Residents Hospitals 1: 2 1 1: (2): 1 3 2 5 6 2: 1 2 2: (2): 2 6 1 4 5 3: 1 3 3: (2): 4 3 4: 2 3 5: 2 1 6: 1 2 • Unacquainted pairs U = {(1,2), (3,1), (5,2)} • Although pairs may block a matching M in theory, there is no guarantee they will block M in practice • If no social ties exist between pairs they are far less likely to form blocking pairs • if they do not know about each other’s preferences and matched partners • Relaxing the stability criteria to consider only pairs that are likely to block a matching in practice forms the Hospitals/Residents problem under Social Stability (HRSS)

Hospitals/Residents Problem under Social Stability (HRSS) Residents Hospitals Residents Hospitals 1: 2 1 1: (2): 1 3 2 5 6 2: 1 2 2: (2): 2 6 1 4 5 3: 1 3 3: (2): 4 3 4: 2 3 5: 2 1 6: 1 2 • Unacquainted pairs U = {(1,2), (3,1), (5,2)} 1 2 1 3 2 4 3 5 6 Social network graph G • An instance (I, G) of HRSS consists of: • A HR instance I • A social network graph G = (R∪H, A) with edges that correspond to pairs that may form blocking pairs with respect to a matching in (I, G). We call such edges acquainted pairs • Unacquainted pairs are acceptable pairs that are not in G • We denote the set of unacquainted pairs as U • Unacquainted pairs cannot block a matching in (I, G)

Hospitals/Residents Problem under Social Stability (HRSS) • We investigate a relaxed stability criteria in the HRSS context – social stability • A pair (r, h)socially blocks a matching Mor forms a social blocking pair with respect to M if and only if • (r, h) blocks M in the classical sense • (r, h) is an acquainted pair • A socially stable matching is one that admits no social blocking pairs • Given HR and HRSS instances I and (I, G) respectively, any stable matching in I is also socially stable in (I, G)

Hospitals/Residents Problem under Social Stability (HRSS) Residents Hospitals Residents Hospitals 1: 2 1 1: (2): 1 3 2 5 6 2: 1 2 2: (2): 2 6 1 4 5 3: 1 3 3: (2): 4 3 4: 2 3 5: 2 1 6: 1 2 • Unacquainted pairs U = {(1,2), (3,1), (5,2)} • Socially Stable Matching M’ = {(1,2), (2,1), (3,3), (4,3), (5,1), (6,2)} |M’| = 6 1 2 1 3 2 4 3 5 6 Social network graph G • An unacquainted pair cannot block a matching even if both agents prefer each other to their matched partners • An instance of HRSS can admit socially stable matchings of varying sizes • Socially stable matchings can be larger than stable matchings • can be twice the size of stable matchings in a given instance

Hospitals/Residents Problem under Social Stability (HRSS) • The one-to-one variant of HRSS is the Stable Marriage with Incomplete lists under Social Stability (SMISS) • When only one set of agents are involved, the problem is called the Stable Roommates Problem with Free pairs (SRF) • MAX HRSS and MAX SMISS are the problems of finding a maximum socially stable matching given instances of HRSS and SMISS respectively • COM HRSS and COM SMISS are the problems of deciding whether a complete socially stable matching exists in HRSS and SMISS instances respectively • (a, b)-MAX SMISS is the restriction of MAX SMISS in which the preference list lengths of men and women are bounded by a and b respectively

Important Results • Previous results: • It is NP-complete to determine if an instance of SRF admits a socially stable matching (Cechlarova and Fleiner ‘09) • A similar model to HRSS was investigated by Cheng and McDermid ’12 (referred to as HR+SN) and Arcaute and Vassilvitskii ’09 • Hoefer and Wagner ‘13 investigated a generalisation of HRSS and HR+SN • Our Contributions: • (3,3)-COM SMISS is NP-complete • (2, ∞)-MAX SMISS can be solved in polynomial time • MAX HRSS is solvable in polynomial time if either the number of acquainted pairs is constant or the number of unacquainted pairs is constant • MAX HRSS can be approximated to within a factor of 3/2

Approximating MAX HRSS • A stable matching in the underlying SMI instance is also socially stable and is a 2-approximation of MAX SMISS • An instance (I, G) of HRSS can be transformed in polynomial time to an instance of (I’, G’) SMISS by cloning • A hospital hiwith p posts is cloned into hi,1, hi,2, …, hi,pcopies • Each clone adopts the preference list of its parent hospital • hi is replaced on each resident’s preference list in which it appeared with all p clones of hi in increasing indicial order on second subscripts • If (r, hi) is in G then (r, hi,x) is in G’ for (1 ≤ x ≤ p) • a socially stable matching M in (I, G) can be transformed in polynomial time to a socially stable matching M’ in (I’, G’) such that |M| = |M’| and vice versa • Thus a c-approximation algorithm for MAX SMISS is also a c-approximation algorithm for MAX HRSS

Approximating MAX SMISS • Men propose to the women on their list and are either accepted or rejected • Unmatched men are promoted and given a second chance to propose (along the lines of Király, 2008) • Women accept or reject proposals based on a preference relation between their currently matched partner and the proposing man which takes into account whether a given man has been promoted and/or forms an edge with her in G • Preference relation is normally based on the position of the men on her list Women 1: 2 1 4 3 2: 4 3 1 2

Approximating MAX SMISS • We modify the preference relation as follows: Women 1: 2 1 4p 3 2: 4 3 1 2 Women 1: 2 1 4 3 2: 4 3 1 2

Approximating MAX SMISS

Approximating MAX SMISS Men Women 1: 3 1 2 1: 1 2 2: 1 2: 1 3: 3 3: 3 5 1 4: 4 5 4: 4 5 5: 3 4 5: 4 • Men 3 and 5 remain unmatched after the first round of proposals. • They are promoted and given another chance

Approximating MAX SMISS Men Women 1: 3 1 2 1: 1 2 2: 1 2: 1 3: 3 3: 3p 5p 1 4: 4 5 4: 4 5p 5: 3 4 5: 4 Mopt = {(1,2), (2,1), (3,3), (4,5), (5,4)} • Algorithm runs in O(n1m) time where n1 is the number of men and m is the total number of acceptable pairs • Algorithm has a performance guarantee of 3/2 (bound is tight)

Future Work • Is there a polynomial time algorithm for (2, ∞)-MAX HRSS? • One may argue that any under-subscribed hospital will form an implicit admissible pair with all residents (as knowledge of its under-subscription may become public)

Thank You

Socially stable matchings in the Hospitals/Residents problem