Student-Project Allocation with Preferences over Projects

1
Student-Project Allocation withPreferences over
Projects

• David Manlove
• University of GlasgowDepartment of Computing
  Science
• Joint work with Gregg OMalley

Supported by EPSRC grant GR/R84597/01,RSE /
Scottish Exec Personal Research Fellowship
2
Background and motivation

• Students may undertake project work during degree
  course
• Set of students, projects and lecturers
• Typically a wide range of projects exceeding
  number of students
• Students may rank projects in preference order
• Lecturers may have preferences over students /
  projects
• Projects / lecturers may have capacities

3
Related work (1)

• Growing interest in automating the allocation
  process
• Efficient algorithms are important
• Formalise the matching problem
• The Student-Project Allocation problem (SPA)
• No explicit lecturer preferences
• University of Southampton
• Proll (1972), bottleneck assignment problem
• Teo and Ho (1998), greedy approach
• Anwar and Bahaj (2003), integer programming
• Harper et al (2005), genetic algorithm

4
Related work (2)

• Lecturer preferences over students
• Project and lecturer capacities equal to 1
• University of York, Department of Computer
  Science
• Constraint programming for stable marriage
  variants
• Dye (2001), Kazakov (2002), Thorn (2003)
• Arbitrary project and lecturer capacities
• Linear-time combinatorial algorithms
• Abraham, Irving and DFM, The student-project
  allocation problem, Proc. ISAAC 2003, LNCS
• Abraham, Irving and DFM, Two algorithms for the
  student-project allocation problem, 2004,
  submitted

5
Lecturer preferencesover projects

• Lecturer preferences over students
• Defaults to academic merit order?
• Weaker students obtain less preferable projects
• Lecturer preferences over projects
• Ranking could reflect research interests, for
  example
• Lecturer implicitly indifferent among all
  students who find a given project acceptable
• Student-Project Allocation problem with Project
  preferences (SPA-P)
• DFM and OMalley, Student-Project Allocation
  with Preferences over Projects, Proc. ACID 2005,
  to appear
• Seek a stable matching as a solution
• Roth (1984)

6
Formal definition of SPA-P

• Set of students Ss1, s2, , sn
• Set of projects Pp1, p2, , pm
• Set of lecturers Ll1, l2, , lq
• Each student si finds acceptable a set of
  projects Ai ? P
• si ranks Ai in strict order of preference
• Each project pj has a capacity cj
• Each lecturer lk has a capacity dk
• Each lecturer lk offers a set of projects Pk ? P
• lk ranks Pk in strict order of preference
• assume that P1, P2, , Pq partitions P

7
Example SPA-P instance

• Student preferences Lecturer preferences Lecturer
  capacities
• s1 p1 p4 p3 l1 p1 p2 p3
  3
• s2 p5 p1 Project capacities 1 2
  1
• s3 p2 p5
• s4 p4 p2 l2 p4 p5 2
• s5 p5 p2 Project capacities 1 2
• Lecturer capacities d1 3, d2 2
• Project capacities c1 1 c2 2 c3 1 c4
  2 c5 1

8
Definition of a matching

• An assignment M is a subset of SP such that if
  (si, pj)?M then si finds pj acceptable

9
Definition of a matching

• An assignment M is a subset of SP such that if
  (si, pj)?M then si finds pj acceptable
• If (si, pj)?M , where lk offers pj , we say that
• si is assigned to pj
• si is assigned to lk
• pj is assigned si
• lk is assigned si

10
Definition of a matching

• An assignment M is a subset of SP such that if
  (si, pj)?M then si finds pj acceptable
• If (si, pj)?M , where lk offers pj , we say that
• si is assigned to pj
• si is assigned to lk
• pj is assigned si
• lk is assigned si
• For any student si , M(si) denotes the set of
  projects that si is assigned to
• For any project pj , M(pj) denotes the set of
  students assigned to pj
• For any lecturer lk , M(lk) denotes the set of
  students assigned to (projects offered by) lk

11
Definition of a matching

• An assignment M is a subset of SP such that if
  (si, pj)?M then si finds pj acceptable
• If (si, pj)?M , where lk offers pj , we say that
• si is assigned to pj
• si is assigned to lk
• pj is assigned si
• lk is assigned si
• For any student si , M(si) denotes the set of
  projects that si is assigned to
• For any project pj , M(pj) denotes the set of
  students assigned to pj
• For any lecturer lk , M(lk) denotes the set of
  students assigned to (projects offered by) lk
• A matching M is an assignment such that
  M(si)?1, M(pj)?cj and M(lk)?dk

12
Example matching

• Student preferences Lecturer preferences Lecturer
  capacities
• s1 p1 p4 p3 l1 p1 p2
  p3 2/3
• s2 p5 p1 Project capacities 0/1 1/2
  1/1
• s3 p2 p5
• s4 p4 p2 l2 p4 p5 2/2
• s5 p5 p2 Project capacities 0/1 2/2
• Lecturer capacities d1 3, d2 2
• Project capacities c1 1 c2 2 c3 1 c4
  1 c5 2

13
Stable matchings

• (si, pj) is a blocking pair of a matching M if
• pj ? Ai
• Either si is unmatched in M, or si prefers pj to
  M(si)
• pj is under-subscribed and either
• si ?M(lk) and lk prefers pj to M(si)
• si ?M(lk) and lk is under-subscribed
• si ?M(lk) and lk prefers pj to his worst
  non-empty project
• where lk is the lecturer who offers pj
• A matching M is stable if it admits no blocking
  pair

14
Example blocking pair (1)

• Student preferences Lecturer preferences Lecturer
  capacities
• s1 p1 p4 p3 l1 p1 p2
  p3 2/3
• s2 p5 p1 Project capacities 0/1 1/2
  1/1
• s3 p2 p5
• s4 p4 p2 l2 p4 p5 2/2
• s5 p5 p2 Project capacities 0/1 2/2
• (s1, p1) is a blocking pair, since
• 3. p1 is under-subscribed and
• s1 ?M(l1) and l1 prefers p1 to M(s1)p3

15
Example blocking pair (2)

• Student preferences Lecturer preferences Lecturer
  capacities
• s1 p1 p4 p3 l1 p1 p2
  p3 2/3
• s2 p5 p1 Project capacities 0/1 1/2
  1/1
• s3 p2 p5
• s4 p4 p2 l2 p4 p5 2/2
• s5 p5 p2 Project capacities 0/1 2/2
• (s2, p1) is a blocking pair, since
• 3. p1 is under-subscribed and
• s2 ?M(l1) and l1 is under-subscribed

16
Example blocking pair (3)

• Student preferences Lecturer preferences Lecturer
  capacities
• s1 p1 p4 p3 l1 p1 p2
  p3 2/3
• s2 p5 p1 Project capacities 0/1 1/2
  1/1
• s3 p2 p5
• s4 p4 p2 l2 p4 p5 2/2
• s5 p5 p2 Project capacities 0/1 2/2
• (s4, p4) is a blocking pair, since
• 3. p4 is under-subscribed and
• s4 ?M(l2) and l2 prefers p4 to his worst
  non-empty project

17
Example stable matching

• Student preferences Lecturer preferences Lecturer
  capacities
• s1 p1 p4 p3 l1 p1 p2
  p3 2/3
• s2 p5 p1 Project capacities 1/1 1/2
  0/1
• s3 p2 p5
• s4 p4 p2 l2 p4 p5 2/2
• s5 p5 p2 Project capacities 1/1 1/2

18
Sizes of stable matchings

• Every instance of SPA-P admits at least one
  stable matching

19
Sizes of stable matchings

• Every instance of SPA-P admits at least one
  stable matching
• But, stable matchings can have different sizes,
  e.g.
• Student preferences Lecturer preferences
• s1 p1 p2 l1 p1
• s2 p1 l2 p2 (each project and lecturer
  has capacity 1)

20
Sizes of stable matchings

• Every instance of SPA-P admits at least one
  stable matching
• But, stable matchings can have different sizes,
  e.g.
• Student preferences Lecturer preferences
• s1 p1 p2 l1 p1
• s2 p1 l2 p2 (each project and lecturer
  has capacity 1)
• M1(s1, p1)

21
Sizes of stable matchings

• Every instance of SPA-P admits at least one
  stable matching
• But, stable matchings can have different sizes,
  e.g.
• Student preferences Lecturer preferences
• s1 p1 p2 l1 p1
• s2 p1 l2 p2 (each project and lecturer
  has capacity 1)
• M1(s1, p1)
• Student preferences Lecturer preferences
• s1 p1 p2 l1 p1
• s2 p1 l2 p2 (each project and lecturer
  has capacity 1)
• M2(s1, p2), (s2, p1)

22
Maximisation problem

• MAX-SPA-P denotes the problem of finding a
  maximum cardinality stable matching, given an
  instance of SPA-P
• ALL-SPA-P denotes the problem of deciding whether
  an instance of SPA-P admits a stable matching in
  which every student is matched
• We show that ALL-SPA-P is NP-complete
• Immediate corollary is that MAX-SPA-P is NP-hard
• Result holds even if each project and lecturer
  has capacity 1

23
Exact Maximal Matching

• A matching M in a graph G is maximal if M?e is
  not a matching for any edge e?M

u1
w1
w2
u2
w3
u3

• The following problem is NP-complete
• EXACT-MM
• Input Bipartite graph G and integer K
• Question does G contain a maximal matching of
  size (exactly) K?

24
Overview of the reduction
must ensure that ui or wj matched
ui
EXACT-MM instance
wj
assume n2 RH vertices
assume n1 LH vertices
25
Overview of the reduction
must ensure that ui or wj matched
ui
EXACT-MM instance
wj
assume n2 RH vertices
assume n1 LH vertices
Student preferences
Lecturer preferences
ALL-SPA-P instance
si pj y1yn1-K
qj pj zj
(1 ? j ? n2)
(1 ? i ? n1)
ti z1zn2
rj yj
(1 ? i ? n2-K)
(1 ? j ? n1-K)
Each project and lecturer has capacity 1
26
Overview of the reduction
must ensure that ui or wj matched
ui
EXACT-MM instance
wj
assume n2 RH vertices
assume n1 LH vertices
Student preferences
Lecturer preferences
ALL-SPA-P instance
si pj y1yn1-K
qj pj zj
(1 ? j ? n2)
(1 ? i ? n1)
ti z1zn2
rj yj
(1 ? i ? n2-K)
(1 ? j ? n1-K)
Each project and lecturer has capacity 1
27
Approximation algorithm
M M ? (si, pj) / si is
provisionally assigned to pj and to lk / if
(lk is full) pz lks worst non-empty
project for (each successor pt of pz on
lks list) for (each student sr such
that sr?At) delete pt from srs
list

• M ?
• while (some student si is free and
• si has a nonempty list)
• pj first project on sis list
• lk lecturer who offers pj
• / si applies to pj /
• if (pj is full)
• delete pj from sis list
• else if (lk is full)
• pz lks worst non-empty project
• if (pz pj )
• delete pj from sis list
• else
• sr some student in M(pz)
• M M \ (sr , pz)
• delete pz from srs list

28
Approximation algorithm
M M ? (si, pj) / si is
provisionally assigned to pj and to lk / if
(lk is full) pz lks worst non-empty
project for (each successor pt of pz on
lks list) for (each student sr such
that sr?At) delete pt from srs
list

• M ?
• while (some student si is free and
• si has a nonempty list)
• pj first project on sis list
• lk lecturer who offers pj
• / si applies to pj /
• if (pj is full)
• delete pj from sis list
• else if (lk is full)
• pz lks worst non-empty project
• if (pz pj )
• delete pj from sis list
• else
• sr some student in M(pz)
• M M \ (sr , pz)
• delete pz from srs list

29
Approximation algorithm
M M ? (si, pj) / si is
provisionally assigned to pj and to lk / if
(lk is full) pz lks worst non-empty
project for (each successor pt of pz on
lks list) for (each student sr such
that sr?At) delete pt from srs
list

• M ?
• while (some student si is free and
• si has a nonempty list)
• pj first project on sis list
• lk lecturer who offers pj
• / si applies to pj /
• if (pj is full)
• delete pj from sis list
• else if (lk is full)
• pz lks worst non-empty project
• if (pz pj )
• delete pj from sis list
• else
• sr some student in M(pz)
• M M \ (sr , pz)
• delete pz from srs list

30
Approximation algorithm
M M ? (si, pj) / si is
provisionally assigned to pj and to lk / if
(lk is full) pz lks worst non-empty
project for (each successor pt of pz on
lks list) for (each student sr such
that sr?At) delete pt from srs
list

• M ?
• while (some student si is free and
• si has a nonempty list)
• pj first project on sis list
• lk lecturer who offers pj
• / si applies to pj /
• if (pj is full)
• delete pj from sis list
• else if (lk is full)
• pz lks worst non-empty project
• if (pz pj )
• delete pj from sis list
• else
• sr some student in M(pz)
• M M \ (sr , pz)
• delete pz from srs list

31
Approximation algorithm
M M ? (si, pj) / si is
provisionally assigned to pj and to lk / if
(lk is full) pz lks worst non-empty
project for (each successor pt of pz on
lks list) for (each student sr such
that sr?At) delete pt from srs
list

• M ?
• while (some student si is free and
• si has a nonempty list)
• pj first project on sis list
• lk lecturer who offers pj
• / si applies to pj /
• if (pj is full)
• delete pj from sis list
• else if (lk is full)
• pz lks worst non-empty project
• if (pz pj )
• delete pj from sis list
• else
• sr some student in M(pz)
• M M \ (sr , pz)
• delete pz from srs list

32
Approximation algorithm
M M ? (si, pj) / si is
provisionally assigned to pj and to lk / if
(lk is full) pz lks worst non-empty
project for (each successor pt of pz on
lks list) for (each student sr such
that sr?At) delete pt from srs
list

• M ?
• while (some student si is free and
• si has a nonempty list)
• pj first project on sis list
• lk lecturer who offers pj
• / si applies to pj /
• if (pj is full)
• delete pj from sis list
• else if (lk is full)
• pz lks worst non-empty project
• if (pz pj )
• delete pj from sis list
• else
• sr some student in M(pz)
• M M \ (sr , pz)
• delete pz from srs list

33
Example SPA-P instance

• Student preferences Lecturer preferences
  Lecturer capacities
• s1 p1 p2 p6 l1 p3 p1 p2
  p4 3
• s2 p1 p4 Project capacities 1 2
  2 1
• s3 p1 p2 p5
• s4 p3 l2 p5 p6 3
• s5 p3 p5 Project capacities 1
  2
• s6 p5 p3 p6
• Lecturer capacities d1 3, d2 3
• Project capacities c1 2 c2 2 c3 1 c4
  1 c5 1 c6 2

34
Example SPA-P instance

• Student preferences Lecturer preferences
  Lecturer capacities
• s1 p1 p2 p6 l1 p3 p1 p2
  p4 1/3
• s2 p1 p4 Project capacities 0/1 1/2
  0/2 0/1
• s3 p1 p2 p5
• s4 p3 l2 p5 p6 0/3
• s5 p3 p5 Project capacities 0/1
  0/2
• s6 p5 p3 p6
• s1 applies to p1
• p1 remains under-subscribed l1 remains
  under-subscribed

35
Example SPA-P instance

• Student preferences Lecturer preferences
  Lecturer capacities
• s1 p1 p2 p6 l1 p3 p1 p2
  p4 2/3
• s2 p1 p4 Project capacities 0/1 2/2
  0/2 0/1
• s3 p1 p2 p5
• s4 p3 l2 p5 p6 0/3
• s5 p3 p5 Project capacities 0/1
  0/2
• s6 p5 p3 p6
• s2 applies to p1
• p1 becomes full l1 remains under-subscribed

36
Example SPA-P instance

• Student preferences Lecturer preferences
  Lecturer capacities
• s1 p1 p2 p6 l1 p3 p1 p2
  p4 2/3
• s2 p1 p4 Project capacities 0/1 2/2
  0/2 0/1
• s3 p1 p2 p5
• s4 p3 l2 p5 p6 0/3
• s5 p3 p5 Project capacities 0/1
  0/2
• s6 p5 p3 p6
• s3 applies to p1
• p1 is already full

37
Example SPA-P instance

• Student preferences Lecturer preferences
  Lecturer capacities
• s1 p1 p2 p6 l1 p3 p1 p2
  p4 2/3
• s2 p1 p4 Project capacities 0/1 2/2
  0/2 0/1
• s3 p1 p2 p5
• s4 p3 l2 p5 p6 0/3
• s5 p3 p5 Project capacities 0/1
  0/2
• s6 p5 p3 p6
• s3 applies to p1
• p1 is already full p1 is deleted from s3s list
  

?
38
Example SPA-P instance

• Student preferences Lecturer preferences
  Lecturer capacities
• s1 p1 p2 p6 l1 p3 p1 p2
  p4 3/3
• s2 p1 p4 Project capacities 0/1 2/2
  1/2 0/1
• s3 p1 p2 p5
• s4 p3 l2 p5 p6 0/3
• s5 p3 p5 Project capacities 0/1
  0/2
• s6 p5 p3 p6
• s3 applies to p2
• p2 remains under-subscribed l1 becomes full

?
39
Example SPA-P instance

• Student preferences Lecturer preferences
  Lecturer capacities
• s1 p1 p2 p6 l1 p3 p1 p2
  p4 3/3
• s2 p1 p4 Project capacities 0/1 2/2
  1/2 0/1
• s3 p1 p2 p5
• s4 p3 l2 p5 p6 0/3
• s5 p3 p5 Project capacities 0/1
  0/2
• s6 p5 p3 p6
• s3 applies to p2
• p2 remains under-subscribed l1 becomes full
  p4 is deleted from s2s list

?
?
40
Example SPA-P instance

• Student preferences Lecturer preferences
  Lecturer capacities
• s1 p1 p2 p6 l1 p3 p1 p2
  p4 3/3
• s2 p1 p4 Project capacities 0/1 2/2
  1/2 0/1
• s3 p1 p2 p5
• s4 p3 l2 p5 p6 0/3
• s5 p3 p5 Project capacities 0/1
  0/2
• s6 p5 p3 p6
• s4 applies to p3
• l1 is already full

?
?
41
Example SPA-P instance

• Student preferences Lecturer preferences
  Lecturer capacities
• s1 p1 p2 p6 l1 p3 p1 p2
  p4 3/3
• s2 p1 p4 Project capacities 1/1 2/2
  0/2 0/1
• s3 p1 p2 p5
• s4 p3 l2 p5 p6 0/3
• s5 p3 p5 Project capacities 0/1
  0/2
• s6 p5 p3 p6
• s4 applies to p3
• l1 is already full s3 is rejected from p2 p3
  becomes full

?
?
?
42
Example SPA-P instance

• Student preferences Lecturer preferences
  Lecturer capacities
• s1 p1 p2 p6 l1 p3 p1 p2
  p4 3/3
• s2 p1 p4 Project capacities 1/1 2/2
  0/2 0/1
• s3 p1 p2 p5
• s4 p3 l2 p5 p6 0/3
• s5 p3 p5 Project capacities 0/1
  0/2
• s6 p5 p3 p6
• s4 applies to p3
• l1 is already full s3 is rejected from p2 s3
  is rejected from p2
• p2 is deleted from s1s list

?
?
?
?
43
Example SPA-P instance

• Student preferences Lecturer preferences
  Lecturer capacities
• s1 p1 p2 p6 l1 p3 p1 p2
  p4 3/3
• s2 p1 p4 Project capacities 1/1 2/2
  0/2 0/1
• s3 p1 p2 p5
• s4 p3 l2 p5 p6 1/3
• s5 p3 p5 Project capacities 1/1
  0/2
• s6 p5 p3 p6
• s3 applies to p5
• p5 becomes full l2 remains under-subscribed

?
?
?
?
44
Example SPA-P instance

• Student preferences Lecturer preferences
  Lecturer capacities
• s1 p1 p2 p6 l1 p3 p1 p2
  p4 3/3
• s2 p1 p4 Project capacities 1/1 2/2
  0/2 0/1
• s3 p1 p2 p5
• s4 p3 l2 p5 p6 1/3
• s5 p3 p5 Project capacities 1/1
  0/2
• s6 p5 p3 p6
• s5 applies to p3
• p3 is already full

?
?
?
?
45
Example SPA-P instance

• Student preferences Lecturer preferences
  Lecturer capacities
• s1 p1 p2 p6 l1 p3 p1 p2
  p4 3/3
• s2 p1 p4 Project capacities 1/1 2/2
  0/2 0/1
• s3 p1 p2 p5
• s4 p3 l2 p5 p6 1/3
• s5 p3 p5 Project capacities 1/1
  0/2
• s6 p5 p3 p6
• s5 applies to p3
• p3 is already full p3 is deleted from s5s list

?
?
?
?
?
46
Example SPA-P instance

• Student preferences Lecturer preferences
  Lecturer capacities
• s1 p1 p2 p6 l1 p3 p1 p2
  p4 3/3
• s2 p1 p4 Project capacities 1/1 2/2
  0/2 0/1
• s3 p1 p2 p5
• s4 p3 l2 p5 p6 1/3
• s5 p3 p5 Project capacities 1/1
  0/2
• s6 p5 p3 p6
• s5 applies to p5
• p5 is already full

?
?
?
?
?
47
Example SPA-P instance

• Student preferences Lecturer preferences
  Lecturer capacities
• s1 p1 p2 p6 l1 p3 p1 p2
  p4 3/3
• s2 p1 p4 Project capacities 1/1 2/2
  0/2 0/1
• s3 p1 p2 p5
• s4 p3 l2 p5 p6 1/3
• s5 p3 p5 Project capacities 1/1
  0/2
• s6 p5 p3 p6
• s5 applies to p5
• p5 is already full p5 is deleted from s5s list

?
?
?
?
?
?
48
Example SPA-P instance

• Student preferences Lecturer preferences
  Lecturer capacities
• s1 p1 p2 p6 l1 p3 p1 p2
  p4 3/3
• s2 p1 p4 Project capacities 1/1 2/2
  0/2 0/1
• s3 p1 p2 p5
• s4 p3 l2 p5 p6 1/3
• s5 p3 p5 Project capacities 1/1
  0/2
• s6 p5 p3 p6
• s6 applies to p5
• p5 is already full

?
?
?
?
?
?
49
Example SPA-P instance

• Student preferences Lecturer preferences
  Lecturer capacities
• s1 p1 p2 p6 l1 p3 p1 p2
  p4 3/3
• s2 p1 p4 Project capacities 1/1 2/2
  0/2 0/1
• s3 p1 p2 p5
• s4 p3 l2 p5 p6 1/3
• s5 p3 p5 Project capacities 1/1
  0/2
• s6 p5 p3 p6
• s6 applies to p5
• p5 is already full p5 is deleted from s6s list

?
?
?
?
?
?
?
50
Example SPA-P instance

• Student preferences Lecturer preferences
  Lecturer capacities
• s1 p1 p2 p6 l1 p3 p1 p2
  p4 3/3
• s2 p1 p4 Project capacities 1/1 2/2
  0/2 0/1
• s3 p1 p2 p5
• s4 p3 l2 p5 p6 1/3
• s5 p3 p5 Project capacities 1/1
  0/2
• s6 p5 p3 p6
• s6 applies to p3
• p3 is already full

?
?
?
?
?
?
?
51
Example SPA-P instance

• Student preferences Lecturer preferences
  Lecturer capacities
• s1 p1 p2 p6 l1 p3 p1 p2
  p4 3/3
• s2 p1 p4 Project capacities 1/1 2/2
  0/2 0/1
• s3 p1 p2 p5
• s4 p3 l2 p5 p6 1/3
• s5 p3 p5 Project capacities 1/1
  0/2
• s6 p5 p3 p6
• s6 applies to p3
• p3 is already full p3 is deleted from s6s list

?
?
?
?
?
?
?
?
52
Example SPA-P instance

• Student preferences Lecturer preferences
  Lecturer capacities
• s1 p1 p2 p6 l1 p3 p1 p2
  p4 3/3
• s2 p1 p4 Project capacities 1/1 2/2
  0/2 0/1
• s3 p1 p2 p5
• s4 p3 l2 p5 p6 2/3
• s5 p3 p5 Project capacities 1/1
  1/2
• s6 p5 p3 p6
• s6 applies to p6
• p6 remains under-subscribed l2 remains
  under-subscribed

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53
Example SPA-P instance

• Student preferences Lecturer preferences
  Lecturer capacities
• s1 p1 p2 p6 l1 p3 p1 p2
  p4 3/3
• s2 p1 p4 Project capacities 1/1 2/2
  0/2 0/1
• s3 p1 p2 p5
• s4 p3 l2 p5 p6 2/3
• s5 p3 p5 Project capacities 1/1
  1/2
• s6 p5 p3 p6

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54
Example SPA-P instance

• Student preferences Lecturer preferences
  Lecturer capacities
• s1 p1 p2 p6 l1 p3 p1 p2
  p4 3/3
• s2 p1 p4 Project capacities 1/1 2/2
  0/2 0/1
• s3 p1 p2 p5
• s4 p3 l2 p5 p6 2/3
• s5 p3 p5 Project capacities 1/1
  1/2
• s6 p5 p3 p6
• Algorithm terminates with a stable matching of
  size 5

55
Example SPA-P instance

• Student preferences Lecturer preferences
  Lecturer capacities
• s1 p1 p2 p6 l1 p3 p1 p2
  p4 3/3
• s2 p1 p4 Project capacities 1/1 2/2
  0/2 0/1
• s3 p1 p2 p5
• s4 p3 l2 p5 p6 3/3
• s5 p3 p5 Project capacities 1/1
  2/2
• s6 p5 p3 p6
• Stable matching of size 6

56
Theoretical results

• Algorithm produces a stable matching, given an
  instance of SPA-P
• So every instance of SPA-P admits a stable
  matching
• Approximation algorithm has a performance
  guarantee of 2
• Algorithm may be implemented to run in O(L) time,
  where L is total length of the students
  preference lists
• The constructed matching admits no
  exchange-blocking coalition

57
A worst-case example
Student preferences Lecturer preferences s1 p1
p2 l1 p1 s2 p1 l2 p2 s3 p3 p4
l3 p3 s4 p3 l4 p4 s2n-1 p2n-1
p2n l2n-1 p2n-1 s2n p2n-1 l2n
p2n Each project and lecturer has capacity 1
58
A worst-case example
Student preferences Lecturer preferences s1 p1
p2 l1 p1 s2 p1 l2 p2 s3 p3 p4
l3 p3 s4 p3 l4 p4 s2n-1 p2n-1
p2n l2n-1 p2n-1 s2n p2n-1 l2n
p2n Each project and lecturer has capacity
1 Students apply to projects in increasing
indicial order M1(s1, p1), (s3, p3), ,
(s2n-1, p2n-1), M1n
59
A worst-case example
Student preferences Lecturer preferences s1 p1
p2 l1 p1 s2 p1 l2 p2 s3 p3 p4
l3 p3 s4 p3 l4 p4 s2n-1 p2n-1
p2n l2n-1 p2n-1 s2n p2n-1 l2n
p2n Each project and lecturer has capacity
1 Students apply to projects in decreasing
indicial order M2(s1, p2), (s2, p1), ,
(s2n-1, p2n), (s2n, p2n-1), M22n
60
Open problems

• Improved approximation algorithm?
• Extend to the case where lecturers have
  preferences over (student, project) pairs
• E.g. l1 (s1, p2) (s2, p2) (s1, p3)
• Ties in the preference lists
• Lower bounds on projects