pareto optimality in house allocation problems
Download
Skip this Video
Download Presentation
Pareto Optimality in House Allocation Problems

Loading in 2 Seconds...

play fullscreen
1 / 28

Pareto Optimality in House Allocation Problems - PowerPoint PPT Presentation


  • 175 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Pareto Optimality in House Allocation Problems' - cato


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
pareto optimality in house allocation problems
David Abraham

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

house allocation problem ha
House Allocation problem (HA)

  • 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

applications
Applications
  • 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
      • Abdulkadiroğlu and Sönmez, 1998
  • Other matching problems:
    • US Naval Academy: students to naval officer positions
      • Roth and Sotomayor, 1990
    • Scottish Executive Teacher Induction Scheme
    • Assigning students to projects
the underlying graph
The underlying graph
  • 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

the underlying graph5
The underlying graph
  • 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

the underlying graph6
The underlying graph
  • 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

pareto optimal matchings
Pareto optimal matchings
  • 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

testing for pareto optimality
Testing for Pareto optimality
  • 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

testing for pareto optimality9
Testing for Pareto optimality
  • 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

testing for pareto optimality10
Testing for Pareto optimality
  • 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

testing for pareto optimality11
Testing for Pareto optimality
  • 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
finding a pareto optimal matching
Finding a Pareto optimal matching
  • 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
    • Abdulkadiroğlu and Sönmez, 1998
  • 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)}

related work
Related work
  • 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
faster algorithm for finding a maximum pareto optimal matching
Faster algorithm for finding 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
    • Enforce trade-in-free property
  • Phase 3 – O(m) time
    • Enforce coalition-free property
    • Extension of Gale’s Top-Trading Cycles (TTC) algorithm
      • Shapley and Scarf, 1974
phase 1
Phase 1
    • 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 116
Phase 1
    • 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 117
Phase 1

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

Mnot trade-in-free

phase 2 outline
Phase 2 outline
  • 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 outline19
Phase 2 outline
  • 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 outline20
Phase 2 outline
  • 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 outline21
Phase 2 outline
  • 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
Phase 2 termination
  • 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
Phase 3 outline
  • 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’’)
slide24
Phase 3 termination
  • 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
Initial property rights
  • 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
Minimum Pareto optimal matchings
  • 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 pareto optimal matchings
Interpolation of Pareto 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
Open problems

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?
ad