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Pareto Optimality in House Allocation Problems - PowerPoint PPT Presentation

David Abraham Computer Science Department Carnegie-Mellon University. Pareto Optimality in House Allocation Problems. Katar í na Cechl á rov á Institute of Mathematics PJ Saf á rik University in Ko š ice. David Manlove Department of Computing Science University of Glasgow. Kurt Mehlhorn

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Computer Science Department

Carnegie-Mellon University

Pareto Optimality in House Allocation Problems

Katarína Cechlárová

Institute of Mathematics

PJ Safárik University in Košice

David Manlove

Department of Computing Science

University of Glasgow

Kurt Mehlhorn

Max-Planck-Institut fűr Informatik

Saarbrűcken

Supported by Royal Society of Edinburgh/Scottish Executive Personal Research Fellowship

and Engineering and Physical Sciences Research Council grant GR/R84597/01

• Set of agents A={a1, a2, …, ar}

• Set of houses H={h1, h2, …, hs}

• Each agent ai has an acceptable set of houses Ai H

• ai ranks Ai in strict order of preference

• Example:

• a1 : h2 h1

• a2 :h3h4 h2

• a3 : h4h3

• a4 : h1h4

• Let n=r+s and let m=total length of preference lists

a1 finds h1 and h2 acceptable

a3 prefers h4 to h3

• House allocation context:

• Large-scale residence exchange in Chinese housing markets

• Yuan, 1996

• Allocation of campus housing in American universities, such as Carnegie-Mellon, Rochester and Stanford

• Other matching problems:

• US Naval Academy: students to naval officer positions

• Roth and Sotomayor, 1990

• Scottish Executive Teacher Induction Scheme

• Assigning students to projects

• Weighted bipartite graph G=(V,E)

• Vertex set V=AH

• Edge set:{ai,hj}E if and only if ai findshjacceptable

• Weight of edge {ai,hj} is rank of hjinai’spreference list

• Example

• a1 : h2 h1

• a2 :h3h4 h2

• a3 : h4h3

• a4 : h1h4

a1

h1

2

1

a2

h2

3

1

a3

2

h3

2

1

1

a4

h4

2

• Weighted bipartite graph G=(V,E)

• Vertex set V=AH

• Edge set:{ai,hj}E if and only if ai findshjacceptable

• Weight of edge {ai,hj} is rank of hjinai’spreference list

• Example

• a1 : h2 h1

• a2 :h3h4h2

• a3 : h4h3

• a4 : h1h4

a1

h1

2

M(a1)=h1

1

a2

h2

3

1

a3

2

h3

2

1

1

M={(a1, h1), (a2, h4), (a3, h3)}

a4

h4

2

• Weighted bipartite graph G=(V,E)

• Vertex set V=AH

• Edge set:{ai,hj}E if and only if ai findshjacceptable

• Weight of edge {ai,hj} is rank of hjinai’spreference list

• Example

• a1 : h2h1

• a2 : h3h4 h2

• a3 : h4h3

• a4 : h1h4

a1

h1

2

1

a2

h2

3

1

a3

2

h3

2

1

1

M={(a1, h2), (a2, h3), (a3, h4), (a4, h1)}

a4

h4

2

• A matchingM1 is Pareto optimal if there is no matching M2 such that:

• Some agent is better off in M2 than in M1

• No agent is worse off in M2 than in M1

• Example

• M1 is not Pareto optimal since a1 and a2 could swap houses – each would be better off

• M2 is Pareto optimal

• a1 : h2 h1

• a2 :h1h2

• a3 : h3

• a1 : h2h1

• a2 : h1h2

• a3 : h3

M1

M2

• A matchingM is maximal if there is no agent a and house h, each unmatched in M, such that a finds h acceptable

• A matching M is trade-in-free if there is no matched agent a and unmatched house h such that a prefers h to M(a)

• A matching M is coalition-free if there is no coalition, i.e. a sequence of matched agents a0 ,a1 ,…,ar-1 such that ai prefers M(ai) toM(ai+1) (0ir-1)

• a1 : h2 h1

• a2 :h3h4 h2

• a3 : h4h3

• a4 : h1h4

• Proposition: M is Pareto optimal if and only if M is maximal, trade-in-free and coalition-free

Mis not maximal due to a3 and h3

• A matchingM is maximal if there is no agent a and house h, each unmatched in M, such that a finds h acceptable

• A matching M is trade-in-free if there is no matched agent a and unmatched house h such that a prefers h to M(a)

• A matching M is coalition-free if there is no coalition, i.e. a sequence of matched agents a0 ,a1 ,…,ar-1 such that ai prefers M(ai) toM(ai+1) (0ir-1)

• a1 : h2 h1

• a2 :h3h4 h2

• a3 : h4h3

• a4 : h1h4

• Proposition: M is Pareto optimal if and only if M is maximal, trade-in-free and coalition-free

Mis not trade-in-free due to a2 and h3

• A matchingM is maximal if there is no agent a and house h, each unmatched in M, such that a finds h acceptable

• A matching M is trade-in-free if there is no matched agent a and unmatched house h such that a prefers h to M(a)

• A matching M is coalition-free if there is no coalition, i.e. a sequence of matched agents a0 ,a1 ,…,ar-1 such that ai prefers M(ai+1) toM(ai) (0ir-1)

• a1 : h2 h1

• a2 :h3h4 h2

• a3 : h4h3

• a4 : h1h4

• Proposition: M is Pareto optimal if and only if M is maximal, trade-in-free and coalition-free

a1

h1

Mis not coalition-free

due to a1, a2, a4

a2

h2

a3

h3

a4

h4

• A matching M is maximal if there is no agent a and house h, each unmatched in M, such that a finds h acceptable

• A matching M is trade-in-free if there is no matched agent a and unmatched house h such that a prefers h to M(a)

• A matching M is coalition-free if there is no coalition, i.e. a sequence of matched agents a0 ,a1 ,…,ar-1 such that ai prefers M(ai+1) toM(ai) (0  i  r-1)

• Lemma:M is Pareto optimal if and only if M is maximal, trade-in-free and coalition-free

• Theorem: we may check whether a given matching M is Pareto optimal in O(m) time

• Simple greedy algorithm, referred to as the serial dictatorship mechanism by economists

for each agent a in turn

if a has an unmatched house on his list

match a to the most-preferred such house;

else

report a as unmatched;

• Theorem: The serial dictatorship mechanism constructs a Pareto optimal matching in O(m) time

• Example

• a1 : h1h2h3

• a2 : h1h2

• a3 : h1h2

M1={(a1,h1), (a2,h2)}

• a1 : h1 h2h3

• a2 : h1h2

• a3 : h1h2

M2={(a1,h3), (a2,h2), (a3,h1)}

• Rank maximal matchings

• Matching M is rank maximal if, in M

• Maximum number of agents obtain their first-choice house;

• Subject to (1), maximum number of agents obtain their second-choice house;

etc.

• Irving, Kavitha, Mehlhorn, Michail, Paluch, SODA 04

• A rank maximal matching is Pareto optimal, but need not be of maximum size

• Popular matchings

• Matching M is popular if there is no other matching M’such that:

• more agents prefer M’ to M than prefer M to M’

• Abraham, Irving, Kavitha, Mehlhorn, SODA 05

• A popular matchingis Pareto optimal, but need not exist

• Maximum cardinality minimum weight matchings

• Such a matching M may be found in G in O(nmlog n) time

• Gabow and Tarjan, 1989

• M is a maximum Pareto optimal matching

• Three-phase algorithm with O(nm) overall complexity

• Phase 1 – O(nm) time

• Find a maximum matching in G

• Classical O(nm) augmenting path algorithm

• Hopcroft and Karp, 1973

• Phase 2 – O(m) time

• Phase 3 – O(m) time

• Enforce coalition-free property

• Extension of Gale’s Top-Trading Cycles (TTC) algorithm

• Shapley and Scarf, 1974

Phase 1 matching

• a1 : h4h5h3h2h1

• a2 : h3h4h5h9h1h2

• a3 : h5h4h1h2h3

• a4 : h3h5h4

• a5 : h4h3h5

• a6 : h2h3h5h8h6h7h1h11h4h10

• a7 : h1h4h3h6h7h2h10h5h11

• a8 : h1h5h4h3h7h6h8

• a9 : h4 h3h5h9

• Maximum matching M in G has size 8

• M must be maximal

• No guarantee that M is trade-in-free or coalition-free

• Phase 1 matching

• a1 : h4h5h3h2h1

• a2 : h3h4h5h9h1h2

• a3 : h5h4h1h2h3

• a4 : h3h5h4

• a5 : h4h3h5

• a6 : h2h3h5h8h6h7h1h11h4h10

• a7 : h1h4h3h6h7h2h10h5h11

• a8 : h1h5h4h3h7h6h8

• a9 : h4 h3h5h9

• Maximum matching M in G has size 9

• M must be maximal

• No guarantee that M is trade-in-free or coalition-free

• Phase 1 matching

Mnot coalition-free

• a1 : h4h5h3h2h1

• a2 : h3h4h5h9h1h2

• a3 : h5h4h1h2h3

• a4 : h3h5h4

• a5 : h4h3h5

• a6 : h2h3h5h8h6h7h1h11h4h10

• a7 : h1h4h3h6h7h2h10h5h11

• a8 : h1h5h4h3h7h6h8

• a9 : h4 h3h5h9

• Maximum matching M in G has size 9

• M must be maximal

• No guarantee that M is trade-in-free or coalition-free

Phase 2 outline matching

• Repeatedly search for a matched agent a and an unmatched house h such that a prefers h to h’=M(a)

• Promote a to h

• h’is now unmatched

• Example

• a1 : h4h5h3h2h1

• a2 : h3h4h5h9h1h2

• a3 : h5h4h1h2h3

• a4 : h3h5h4

• a5 : h4h3h5

• a6 : h2h3h5h8h6h7h1h11h4h10

• a7 : h1h4h3h6h7h2h10h5h11

• a8 : h1h5h4h3h7h6h8

• a9 : h4 h3h5h9

Phase 2 outline matching

• Repeatedly search for a matched agent a and an unmatched house h such that a prefers h to h’=M(a)

• Promote a to h

• h’is now unmatched

• Example

• a1 : h4h5h3h2h1

• a2 : h3h4h5h9h1h2

• a3 : h5h4h1h2h3

• a4 : h3h5h4

• a5 : h4h3h5

• a6 : h2h3h5h8h6h7h1h11h4h10

• a7 : h1h4h3h6h7h2h10h5h11

• a8 : h1h5h4h3h7h6h8

• a9 : h4 h3h5h9

Phase 2 outline matching

• Repeatedly search for a matched agent a and an unmatched house h such that a prefers h to h’=M(a)

• Promote a to h

• h’is now unmatched

• Example

• a1 : h4h5h3h2h1

• a2 : h3h4h5h9h1h2

• a3 : h5h4h1h2h3

• a4 : h3h5h4

• a5 : h4h3h5

• a6 : h2h3h5h8h6h7h1h11h4h10

• a7 : h1h4h3h6h7h2h10h5h11

• a8 : h1h5h4h3h7h6h8

• a9 : h4 h3h5h9

Phase 2 outline matching

• Repeatedly search for a matched agent a and an unmatched house h such that a prefers h to h’=M(a)

• Promote a to h

• h’is now unmatched

• Example

• a1 : h4h5h3h2h1

• a2 : h3h4h5h9h1h2

• a3 : h5h4h1h2h3

• a4 : h3h5h4

• a5 : h4h3h5

• a6 : h2h3h5h8h6h7h1h11h4h10

• a7 : h1h4h3h6h7h2h10h5h11

• a8 : h1h5h4h3h7h6h8

• a9 : h4 h3h5h9

Phase 2 termination matching

• Once Phase 2 terminates, matching is trade-in-free

• With suitable data structures, Phase 2 is O(m)

• Coalitions may remain…

• a1 : h4h5h3h2h1

• a2 : h3h4h5h9h1h2

• a3 : h5h4h1h2h3

• a4 : h3h5h4

• a5 : h4h3h5

• a6 : h2h3h5h8h6h7h1h11h4h10

• a7 : h1h4h3h6h7h2h10h5h11

• a8 : h1h5h4h3h7h6h8

• a9 : h4 h3h5h9

Phase 3 outline matching

• Build a path P of agents (represented by a stack)

• Each house is initially unlabelled

• Each agent a has a pointer p(a) pointing to M(a) or the first unlabelled house on a’s preference list (whichever comes first)

• Keep a counter c(a) for each agent a (initially c(a)=0)

• This represents the number of times a appears on the stack

• Outer loop iterates over each matched agent a such that p(a)M(a)

• Initialise P to contain agent a

• Inner loop iterates while P is nonempty

• Pop an agent a’ from P

• If c(a’)=2 we have a coalition (CYCLE)

• Remove by popping the stack and label the houses involved

• Else if p(a’)=M(a’) we reach a dead end (BACKTRACK)

• Label M(a’)

• Else add a’’ wherep(a’)=M(a’’)to the path (EXTEND)

• Push a’ and a’’ onto the stack

• Increment c(a’’)

Phase 3 termination matching

• Once Phase 3 terminates, matching is coalition-free

• a1 : h4h5h3h2h1

• a2 : h3h4h5h9h1h2

• a3 : h5h4h1h2h3

• a4 : h3h5h4

• a5: h4h3h5

• a6 : h2h3h5h8h6h7h1h11h4h10

• a7 : h1h4h3h6h7h2h10h5h11

• a8 : h1h5h4h3h7h6h8

• a9 : h4 h3h5h9

• Phase 3 is O(m)

• Theorem:A maximum Pareto optimal matching can be found in O(nm) time

Initial property rights matching

• Suppose A’A and each member of A’ owns a house initially

• For each agent aA’,denote this house by h(a)

• Truncate a’s list at h(a)

• Form matching M by pre-assigning a to h(a)

• Use Hopcroft-Karp algorithm to augment M to a maximum cardinality matching M’ in restricted HA instance

• Then proceed with Phases 2 and 3 as before

• Constructed matching M’ is individually rational

• If A’=A then we have a housing market

• TTC algorithm finds the unique matching that belongs to the core

• Shapley and Scarf, 1974

• Roth and Postlewaite, 1977

Minimum Pareto optimal matchings matching

• Theorem: Problem of finding a minimum Pareto optimal matching is NP-hard

• Result holds even if all preference lists have length 3

• Reduction from Minimum Maximal Matching

• Problem is approximable within a factor of 2

• Follows since any Pareto optimal matching is a maximal matching in the underlying graph G

• Any two maximal matchings differ in size by at most a factor of 2

• Korte and Hausmann, 1978

Interpolation of matchingPareto optimal matchings

• Given an HA instance I, p-(I) and p+(I) denote the sizes of a minimum and maximum Pareto optimal matching

• Theorem: I admits a Pareto optimal matching of size k, for each k such that p-(I) k p+(I)

• Given a Pareto optimal matching of size k, O(m) algorithm constructs a Pareto optimal matching of size k+1 or reports that k=p+(I)

• Based on assigning a vector r1,…,rk to an augmenting path P=a1,h1,…,ak,hk where ri=rankai(hi)

• Examples: 1,3,21,2,2

• Find a lexicographically smallest augmenting path

1

a1

h1

1

a2

h2

2

3

1

a3

h3

2

1

a4

h4

2

h5

Open problems matching

Finding a maximum Pareto optimal matching

• Ties in the preference lists

• Solvable in O(nmlog n) time

• Solvable in O(nm) time?

• One-many case (houses may have capacity >1)

• Non-bipartite case

• Solvable in O((n(m, n))mlog3/2 n) time

• D.J. Abraham, D.F. Manlove

Pareto optimality in the Roommates problem

Technical Report TR-2004-182 of the Computing Science Department of Glasgow University

• Solvable in O(nm) time?