Algorithms%20for%20Student-Project%20Allocation

1
Algorithms for Student-Project Allocation

• David Manlove
• University of GlasgowDepartment of Computing
  Science
• Joint work with David Abraham and Rob Irving

Supported by EPSRC grant GR/R84597/01,Nuffield
Foundation award NUF-NAL-02, and RSE / SEETLLD
Personal Research Fellowship
2
Background

• 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 rank students in preference order
• Projects / lecturers may have capacities

3
Efficient algorithms

• Growing interest in automating the allocation
  process
• Efficient algorithms are important
• Can identify a family of matching problems
• Range of optimisation criteria possible
• Cases considered

Preferences Preferences Capacities Capacities
Case Students Lecturers Projects Lecturers
1 ? ? ? ?
2 ? ? ? ?
3 ? ? ? ?
4 ? ? ? ?
4
Case 1 formal definition

• No explicit preferences project capacities only
• Set of students Ss1, s2, , sn
• Set of projects Pp1, p2, , pm
• Set of lecturers Ll1, l2, , lq
• Each lecturer lk offers a set of projects Pk ? P
• assume that P1, P2, , Pq partitions P
• Each project pj has a capacity cj
• Each student si finds acceptable a set of
  projects Ai ? P

5
Definition of a matching

• An assignment M is a subset of SP
• 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
• A matching M is an assignment such that
• if si is assigned to pj in M then si finds pj
  acceptable
• si is assigned to at most one project in M
• pj is assigned at most cj students in M

6
Case 1 example

• Set of students, projects and lecturers
• Ss1, s2, s3, s4, s5, Pp1, p2, p3, p4, p5,
  Ll1, l2, l3
• Lecturers offer projects as follows
• P1p1, p2, P2p3, P3p4, p5
• Project capacities
• c1 c2 c4 c5 1, c3 2
• Students find projects acceptable as follows
• A1p1, p3
• A2p1 , p4
• A3p2, p4
• A4p1, p2, p4
• A5p2, p4

7
Corresponding bipartite graph

• capacity

s1
p1
1
s2
p2
1
s3
p3
2
s4
p4
1
p5
s5
1
8
A matching

• capacity

s1
p1
1
s2
p2
1
s3
p3
2
s4
p4
1
p5
s5
1

• A set of edges M is a matching in G if
• each student si is incident to at most one edge
  of M
• each project pj is incident to at most cj edges
  of M

9
A maximum matching

• capacity

s1
p1
1
s2
p2
1
s3
p3
2
s4
p4
1
p5
s5
1
Degree-constrained subgraph problem A maximum
matching may be found in time O where L is the
number of edges (Gabow, 1983)
10
Case 2 student preferences

• Assume that each student si has a
  strictly-ordered ranking list Ri over Ai
• Project capacities only
• Let r denote the maximum length of any students
  preference list
• Define the signature of a matching M to be an
  r-tuple ?x1, x2, , xr? where xi denotes
  the number of students assigned in M to their
  ith-choice project
• A matching M is greedy if M has maximum signature
  (with respect to lexicographic order)
• i.e. the maximum number of students obtain their
  first-choice project, and subject to this
  condition, the maximum number of students obtain
  their second-choice project, etc.

11
Case 2 example

• Set of students, projects and lecturers
• Ss1, s2, s3, Pp1, p2, p3, Ll1
• Project capacities are all 1
• Student preference lists
• s1 p1
• s2 p2 p1 p3
• s3 p2 p3
• Matching M1(s1, p1) (s2, p3) (s3, p2),
  signature ?2, 0, 1?
• Matching M2(s1, p1) (s2, p2) (s3, p3),
  signature ?2, 1, 0?
• M2 is greedy

12
Case 2 example

• Set of students, projects and lecturers
• Ss1, s2, s3, Pp1, p2, p3, Ll1
• Project capacities are all 1
• Student preference lists
• s1 p1
• s2 p2 p1 p3
• s3 p2 p3
• Matching M1(s1, p1), (s2, p3), (s3, p2),
  signature ?2, 0, 1?
• Matching M2(s1, p1) (s2, p2) (s3, p3),
  signature ?2, 1, 0?
• M2 is greedy

13
Case 2 example

• Set of students, projects and lecturers
• Ss1, s2, s3, Pp1, p2, p3, Ll1
• Project capacities are all 1
• Student preference lists
• s1 p1
• s2 p2 p1 p3
• s3 p2 p3
• Matching M1(s1, p1), (s2, p3), (s3, p2),
  signature ?2, 0, 1?
• Matching M2(s1, p1), (s2, p2), (s3, p3),
  signature ?2, 1, 0?
• M2 is greedy

14
Finding a greedy matching

• Traditional method transform to instance of
  assignment problem
• Create weighted bipartite graph G(V, E) with
  VS?P and E(si, pj) pj?Ai
• For any student si and project pj ?Ai, define
  ranki(pj)k, where pj is the kth-choice project
  of si
• Weight of edge (si, pj) is nr-k where kranki(pj)
• Compute a maximum weight matching in G
• Complexity of algorithm O(rn(Lnlog n))
  (Fredman and Tarjan, 1987)
• Two problems
• Arithmetic operations involving edge weights O(r)
• Possible implementation difficulties

15
A direct algorithm

• Combinatorial algorithm O(min(nR, R?n)m) where
  R is the largest rank used in a greedy matching
  (Irving, Kavitha, Mehlhorn, Michail and
  Paluch, Rank-Maximal Matchings, Proc. SODA
  2004)
• Algorithm can be generalised to deal with
  arbitrary project capacities

16
Greedy matchings vs maximum matchings

• Greedy matchings need not be maximum matchings
• E.g. set of students, projects and lecturers
• Ss1, s2, s3, Pp1, p2, p3, Ll1
• Project capacities are all 1
• Student preference lists
• s1 p1 p3
• s2 p2
• s3 p2 p1

17
Greedy matchings vs maximum matchings

• Greedy matchings need not be maximum matchings
• E.g. set of students, projects and lecturers
• Ss1, s2, s3, Pp1, p2, p3, Ll1
• Project capacities are all 1
• Student preference lists
• s1 p1 p3
• s2 p2
• s3 p2 p1
• Two greedy matchings
• M1(s1, p1), (s2, p2), signature ?2, 0?
• M2(s1, p1), (s3, p2), signature ?2, 0?

18
Greedy matchings vs maximum matchings

• Greedy matchings need not be maximum matchings
• E.g. set of students, projects and lecturers
• Ss1, s2, s3, Pp1, p2, p3, Ll1
• Project capacities are all 1
• Student preference lists
• s1 p1 p3
• s2 p2
• s3 p2 p1
• Two greedy matchings
• M1(s1, p1), (s2, p2), signature ?2, 0?
• M2(s1, p1), (s3, p2), signature ?2, 0?
• Maximum matching
• M3(s1, p3), (s2, p2), (s3, p1), signature ?1,
  2?

19
Greedy maximum matchings and rank-minimum
matchings

• Define a greedy maximum matching to be a matching
  with maximum signature, taken over all maximum
  (cardinality) matchings
• The existence of a direct (combinatorial)
  algorithm for finding a greedy maximum matching
  remains open
• Greedy matchings could leave some people very
  badly off
• Alternative notion of optimality define the cost
  of a matching M to be
• cost(M)?ranki(pj) (si, pj)?M
• Define a rank-minimum matching to be a matching
  with minimum cost, taken over all maximum
  matchings

20
Greedy matchings vsrank-minimum matchings

• E.g. set of students, projects and lecturers
• Ss1, s2, s3,, s4, s5, Pp1, p2, p3, p4, p5,
  Ll1
• Project capacities are all 1
• Student preference lists
• s1 p1
• s2 p2 p5
• s3 p3
• s4 p4
• s5 p1 p2 p3 p4 p5
• Greedy matching
• M1(s1, p1), (s2, p2), (s3, p3), (s4, p4), (s5,
  p5), signature ?4, 0, 0, 0, 1?,
  cost 9
• Rank-minimum matching
• M2(s1, p1), (s2, p5), (s3, p3), (s4, p4), (s5,
  p2), signature ?3, 2, 0, 0, 0?,
  cost 7

21
Greedy matchings vsrank-minimum matchings

• E.g. set of students, projects and lecturers
• Ss1, s2, s3,, s4, s5, Pp1, p2, p3, p4, p5,
  Ll1
• Project capacities are all 1
• Student preference lists
• s1 p1
• s2 p2 p5
• s3 p3
• s4 p4
• s5 p1 p2 p3 p4 p5
• Greedy matching
• M1(s1, p1), (s2, p2), (s3, p3), (s4, p4), (s5,
  p5), signature ?4, 0, 0, 0, 1?,
  cost 9
• Rank-minimum matching
• M2(s1, p1), (s2, p5), (s3, p3), (s4, p4), (s5,
  p2), signature ?3, 2, 0, 0, 0?,
  cost 7

22
Greedy matchings vsrank-minimum matchings

• E.g. set of students, projects and lecturers
• Ss1, s2, s3,, s4, s5, Pp1, p2, p3, p4, p5,
  Ll1
• Project capacities are all 1
• Student preference lists
• s1 p1
• s2 p2 p5
• s3 p3
• s4 p4
• s5 p1 p2 p3 p4 p5
• Greedy matching
• M1(s1, p1), (s2, p2), (s3, p3), (s4, p4), (s5,
  p5), signature ?4, 0, 0, 0, 1?,
  cost 9
• Rank-minimum matching
• M2(s1, p1), (s2, p5), (s3, p3), (s4, p4), (s5,
  p2), signature ?3, 2, 0, 0, 0?,
  cost 7

23
Case 3 lecturer capacities

• Assume that each lecturer lk has a capacity dk
• Assume that each student si ranks Ai as before
• Projects have capacities as before
• A matching M is an assignment such that
• if si is assigned to pj in M then si finds pj
  acceptable
• si is assigned to at most one project in M
• pj is assigned at most cj students in M
• lk is assigned at most dk students in M
• An optimal solution is a rank-minimum matching

24
Finding a rank-minimum matching (1)

• Define a weighted network N as follows
• Vertices are Vs?S?P?L?t (s is source and t
  is sink)
• Add an edge (s, si) of capacity 1 for each si
• Add an edge (si, pj) of capacity 1 for each si
  and pj ?Ai
• Add an edge (pj, lk) of capacity cj for each lk
  and pj ?Pk
• Add an edge (lk, t) of capacity dk for each lk
• Edge (si, pj) has cost ranki(pj)
• All other edges have cost 0

25
Finding a rank-minimum matching (2)

• The cost of a flow f is cost(f)?cost(e)f(e)
  e?E
• Find a min cost-max flow f in N
• Using f, create an assignment M as follows
• For each si and pj , if f(si, pj)1 add (si, pj )
  to M
• M is a rank-minimum matching
• Complexity of algorithm O(e(evlog v)log
  B/(mn)) where v is number of
  vertices, e is number of edges and B is sum of
  edge capacities in N (Goldfarb and Jin, 1999)

26
Case 3 example

• Set of students, projects and lecturers
• Ss1, , s5, Pp1, , p7, Ll1, l2, l3
• Lecturers offer projects as follows
• P1p1, p2, P2p3, p4, P3p5, p6, p7
• Project capacities
• p5 has capacity 2 all others have capacity 1
• Lecturer capacities
• d12, d21, d32
• Student preference lists
• s1 p1 p3 p5
• s2 p1 p4 p6
• s3 p4 p1 p5
• s4 p1 p6 p7
• s5 p5 p3 p2

27
Example network N
Edge costs in green Edge capacities in red
p1
1
1
s1
p2
1
1
s2
p3
l1
1
2
1
s
1
1
t
1
s3
p4
l2
1
2
2
1
s4
p5
l3
1
2
3
p6
1
s5
All these edges have capacity 1
p7
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Min cost-max flow f in N
p1
s1
p2
s2
p3
l1
2
s
t
1
s3
p4
l2
2
s4
p5
l3
p6
s5
cost(f)10
Blue edges have flow gt0 - flow is 1 unless stated
otherwise
p7
29
Case 4student and lecturer preferences

• Assume that each student si has a
  strictly-ordered ranking list Ri over Ai
• For each lecturer lk , let Bk denote the set of
  students who find acceptable a project offered by
  lk
• lk has a strictly-ordered ranking list over Bk
• A solution is a stable matching
• A matching M is stable if it admits no blocking
  pair
• Formal definition to follow
• For any student si matched in M, M(si) denotes
  the project 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 lk

30
Definition of a blocking pair

• (si, pj)?M is a blocking pair of M if
• pj ? Ai
• Either si is unmatched in M, or si prefers pj to
  M(si)
• Either
• pj is under-subscribed and lk is
  under-subscribed
• pj is under-subscribed and lk is full, and
  either si?M(lk) or lk prefers si to the worst
  student in M(lk)
• pj is full and lk prefers si to the worst
  student in M(pj)
• where lk is the lecturer who offers pj
• Given this preference and capacity information,
  the problem of finding a stable matching is
  called the Student-Project Allocation Problem
  (SPA)

31
Example SPA instance

• Student preferences Lecturer preferences
• s1 p1 p7 l1 s7 s4 s1 s3 s2 s5
  s6 d1 3
• s2 p1 p2 p3 p4 p5 p6
  l1 offers p1, p2,
  p3
• s3 p2 p1 p4
• s4 p2 l2 s3 s2 s6 s7 s5 d2 2
• s5 p1 p2 p3 p4 l2 offers p4,
  p5, p6
• s6 p2 p3 p4 p5 p6
• s7 p5 p3 p8 l3 s1 s7 d3 2
• l3 offers p7, p8
• c1 2 all other projects have capacity 1

32
SPA instance blocking pair

• Student preferences Lecturer preferences
• s1 p1 p7 l1 s7 s4 s1 s3 s2 s5
  s6 d1 3
• s2 p1 p2 p3 p4 p5 p6
  p2 p1 l1 offers
  p1, p2, p3
• s3 p2 p1 p4
• s4 p2 l2 s3 s2 s6 s7 s5 d2 2
• s5 p1 p2 p3 p4 p4
  p6 l2 offers p4, p5, p6
• s6 p2 p3 p4 p5 p6
• s7 p5 p3 p8 l3 s1 s7 d3 2
• p7 p8 l3 offers p7, p8
• c1 2 all other projects have capacity 1
• (s1, p1) forms a blocking pair
• s1 prefers p1 to M(s1)p7
• p1 is under-subscribed and l1 is under-subscribed

33
SPA instance blocking pair

• Student preferences Lecturer preferences
• s1 p1 p7 l1 s7 s4 s1 s3 s2 s5
  s6 d1 3
• s2 p1 p2 p3 p4 p5 p6
  p2 p1 l1 offers
  p1, p2, p3
• s3 p2 p1 p4
• s4 p2 l2 s3 s2 s6 s7 s5 d2 2
• s5 p1 p2 p3 p4 p4
  p6 l2 offers p4, p5, p6
• s6 p2 p3 p4 p5 p6
• s7 p5 p3 p8 l3 s1 s7 d3 2
• p7 p8 l3 offers p7, p8
• c1 2 all other projects have capacity 1
• (s6, p5) forms a blocking pair
• s6 prefers p5 to M(s6)p6
• p5 is under-subscribed and s6 is assigned to a
  project offered by l2

34
SPA instance blocking pair

• Student preferences Lecturer preferences
• s1 p1 p7 l1 s7 s4 s1 s3 s2 s5
  s6 d1 3
• s2 p1 p2 p3 p4 p5 p6
  p2 p1 p3 l1
  offers p1, p2, p3
• s3 p2 p1 p4
• s4 p2 l2 s3 s2 s6 s7 s5 d2 2
• s5 p1 p2 p3 p4 p4
  p5 l2 offers p4, p5, p6
• s6 p2 p3 p4 p5 p6
• s7 p5 p3 p8 l3 s1 s7 d3 2
• p7 l3 offers p7, p8
• c1 2 all other projects have capacity 1
• (s1, p1) forms a blocking pair
• s1 prefers p1 to M(s1)p7
• p1 is under-subscribed and l1 is full and l1
  prefers s1 to the worst student assigned to l1

35
SPA instance blocking pair

• Student preferences Lecturer preferences
• s1 p1 p7 l1 s7 s4 s1 s3 s2 s5
  s6 d1 3
• s2 p1 p2 p3 p4 p5 p6
  p2 p1 p3 l1
  offers p1, p2, p3
• s3 p2 p1 p4
• s4 p2 l2 s3 s2 s6 s7 s5 d2 2
• s5 p1 p2 p3 p4 p4
  p5 l2 offers p4, p5, p6
• s6 p2 p3 p4 p5 p6
• s7 p5 p3 p8 l3 s1 s7 d3 2
• p7 l3 offers p7, p8
• c1 2 all other projects have capacity 1
• (s5, p3) forms a blocking pair
• s5 is unmatched and finds p3 acceptable
• p3 is full and l1 prefers s5 to the worst student
  assigned to p3

36
SPA instance stable matching

• Student preferences Lecturer preferences
• s1 p1 p7 l1 s7 s4 s1 s3 s2 s5
  s6 d1 3
• s2 p1 p2 p3 p4 p5 p6 p3 p2
  p1 l1 offers p1, p2, p3
• s3 p2 p1 p4
• s4 p2 l2 s3 s2 s6 s7 s5 d2 2
• s5 p1 p2 p3 p4 p4 p5 l2
  offers p4, p5, p6
• s6 p2 p3 p4 p5 p6
• s7 p5 p3 p8 l3 s1 s7 d3 2
• l3 offers p7, p8
• c1 2 all other projects have capacity 1
• The matching is stable

37
Finding a stable matching

• Every instance of SPA admits at least one stable
  matching
• Previous stable matching was student-optimal
• Each student obtains the best project that he/she
  could obtain in any stable matching
• HR is a special case of SPA in which all
  lecturers have capacity ?
• Linear-time algorithms for HR produce stable
  matchings that are student-optimal or
  lecturer-optimal (Gusfield and
  Irving, 1989)
• For SPA with arbitrary lecturer capacities, there
  are also linear-time algorithms giving
  student-optimal and lecturer-optimal stable
  matchings
  (Abraham, Irving, Manlove, The Student-Project
  Allocation Problem, Proc. ISAAC 2003, LNCS vol
  2906)

38
Execution of the algorithm

• Student preferences Lecturer preferences
• s1 p1 p7 l1 s7 s4 s1 s3 s2 s5
  s6 d1 3
• s2 p1 p2 p3 p4 p5 p6 l1
  offers p1, p2, p3
• s3 p2 p1 p4
• s4 p2 l2 s3 s2 s6 s7 s5 d2 2
• s5 p1 p2 p3 p4 l2 offers p4,
  p5, p6
• s6 p2 p3 p4 p5 p6
• s7 p5 p3 p8 l3 s1 s7 d3 2
• l3 offers p7, p8
• c1 2 all other projects have capacity 1

39
Execution of the algorithm

• Student preferences Lecturer preferences
• s1 p1 p7 l1 s7 s4 s1 s3 s2 s5
  s6 d1 3
• s2 p1 p2 p3 p4 p5 p6
  p1 l1 offers p1, p2, p3
• s3 p2 p1 p4
• s4 p2 l2 s3 s2 s6 s7 s5 d2 2
• s5 p1 p2 p3 p4 l2 offers p4,
  p5, p6
• s6 p2 p3 p4 p5 p6
• s7 p5 p3 p8 l3 s1 s7 d3 2
• l3 offers p7, p8
• c1 2 all other projects have capacity 1
• s1 applies to p1
• p1 remains under-subscribed
• l1 remains under-subscribed

40
Execution of the algorithm

• Student preferences Lecturer preferences
• s1 p1 p7 l1 s7 s4 s1 s3 s2 s5
  s6 d1 3
• s2 p1 p2 p3 p4 p5 p6
  p1 p1 l1 offers p1, p2, p3
• s3 p2 p1 p4
• s4 p2 l2 s3 s2 s6 s7 s5 d2 2
• s5 p1 p2 p3 p4 l2 offers p4,
  p5, p6
• s6 p2 p3 p4 p5 p6
• s7 p5 p3 p8 l3 s1 s7 d3 2
• l3 offers p7, p8
• c1 2 all other projects have capacity 1
• s2 applies to p1
• p1 becomes full p1 deleted from list of s5
• l1 remains under-subscribed

?
41
Execution of the algorithm

• Student preferences Lecturer preferences
• s1 p1 p7 l1 s7 s4 s1 s3 s2 s5
  s6 d1 3
• s2 p1 p2 p3 p4 p5 p6
  p1 p2 p1 l1 offers p1, p2, p3
• s3 p2 p1 p4
• s4 p2 l2 s3 s2 s6 s7 s5 d2 2
• s5 p1 p2 p3 p4 l2 offers p4,
  p5, p6
• s6 p2 p3 p4 p5 p6
• s7 p5 p3 p8 l3 s1 s7 d3 2
• l3 offers p7, p8
• c1 2 all other projects have capacity 1
• s3 applies to p2
• p2 becomes full p2 deleted from lists of s2, s5
  and s6
• l1 becomes full p3 deleted from lists of s5 and
  s6

?
?
?
?
?
?
42
Execution of the algorithm

• Student preferences Lecturer preferences
• s1 p1 p7 l1 s7 s4 s1 s3 s2 s5
  s6 d1 3
• s2 p1 p2 p3 p4 p5 p6
  p2 p1 p2 p1 l1 offers p1, p2, p3
• s3 p2 p1 p4
• s4 p2 l2 s3 s2 s6 s7 s5 d2 2
• s5 p1 p2 p3 p4 l2 offers p4,
  p5, p6
• s6 p2 p3 p4 p5 p6
• s7 p5 p3 p8 l3 s1 s7 d3 2
• l3 offers p7, p8
• c1 2 all other projects have capacity 1
• s4 applies to p2
• p2 becomes over-subscribed p2 rejects s3

?
?
?
?
?
?
43
Execution of the algorithm

• Student preferences Lecturer preferences
• s1 p1 p7 l1 s7 s4 s1 s3 s2 s5
  s6 d1 3
• s2 p1 p2 p3 p4 p5 p6
  p2 p1 p1 l1 offers p1, p2, p3
• s3 p2 p1 p4
• s4 p2 l2 s3 s2 s6 s7 s5 d2 2
• s5 p1 p2 p3 p4 l2 offers p4,
  p5, p6
• s6 p2 p3 p4 p5 p6
• s7 p5 p3 p8 l3 s1 s7 d3 2
• l3 offers p7, p8
• c1 2 all other projects have capacity 1
• s4 applies to p2
• p2 becomes over-subscribed p2 rejects s3
• p2 becomes full p2 deleted from list of s3
• l1 remains full

?
?
?
?
?
?
?
44
Execution of the algorithm

• Student preferences Lecturer preferences
• s1 p1 p7 l1 s7 s4 s1 s3 s2 s5
  s6 d1 3
• s2 p1 p2 p3 p4 p5 p6
  p2 p1 p1 p1 l1 offers p1, p2, p3
• s3 p2 p1 p4
• s4 p2 l2 s3 s2 s6 s7 s5 d2 2
• s5 p1 p2 p3 p4 l2 offers p4,
  p5, p6
• s6 p2 p3 p4 p5 p6
• s7 p5 p3 p8 l3 s1 s7 d3 2
• l3 offers p7, p8
• c1 2 all other projects have capacity 1
• s3 applies to p1
• p1 becomes over-subscribed p1 rejects s2

?
?
?
?
?
?
?
45
Execution of the algorithm

• Student preferences Lecturer preferences
• s1 p1 p7 l1 s7 s4 s1 s3 s2 s5
  s6 d1 3
• s2 p1 p2 p3 p4 p5 p6
  p2 p1 p1 l1 offers p1, p2, p3
• s3 p2 p1 p4
• s4 p2 l2 s3 s2 s6 s7 s5 d2 2
• s5 p1 p2 p3 p4 l2 offers p4,
  p5, p6
• s6 p2 p3 p4 p5 p6
• s7 p5 p3 p8 l3 s1 s7 d3 2
• l3 offers p7, p8
• c1 2 all other projects have capacity 1
• s3 applies to p1
• p1 becomes over-subscribed p1 rejects s2
• p1 becomes full p1 deleted from list of s2
• l1 remains full p3 deleted from list of s2

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46
Execution of the algorithm

• Student preferences Lecturer preferences
• s1 p1 p7 l1 s7 s4 s1 s3 s2 s5
  s6 d1 3
• s2 p1 p2 p3 p4 p5 p6
  p2 p1 p1 l1 offers p1, p2, p3
• s3 p2 p1 p4
• s4 p2 l2 s3 s2 s6 s7 s5 d2 2
• s5 p1 p2 p3 p4
  p4 l2 offers p4, p5, p6
• s6 p2 p3 p4 p5 p6
• s7 p5 p3 p8 l3 s1 s7 d3 2
• l3 offers p7, p8
• c1 2 all other projects have capacity 1
• s2 applies to p4
• p4 becomes full p4 deleted from list of s5 and
  s6
• l2 remains under-subscribed

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47
Execution of the algorithm

• Student preferences Lecturer preferences
• s1 p1 p7 l1 s7 s4 s1 s3 s2 s5
  s6 d1 3
• s2 p1 p2 p3 p4 p5 p6
  p2 p1 p1 l1 offers p1, p2, p3
• s3 p2 p1 p4
• s4 p2 l2 s3 s2 s6 s7 s5 d2 2
• s5 p1 p2 p3 p4
  p4 p5 l2 offers p4,
  p5, p6
• s6 p2 p3 p4 p5 p6
• s7 p5 p3 p8 l3 s1 s7 d3 2
• l3 offers p7, p8
• c1 2 all other projects have capacity 1
• s6 applies to p5
• p5 becomes full p5 deleted from list of s7
• l2 becomes full

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48
Execution of the algorithm

• Student preferences Lecturer preferences
• s1 p1 p7 l1 s7 s4 s1 s3 s2 s5
  s6 d1 3
• s2 p1 p2 p3 p4 p5 p6
  p3 p2 p1 p1 l1 offers p1, p2, p3
• s3 p2 p1 p4
• s4 p2 l2 s3 s2 s6 s7 s5 d2 2
• s5 p1 p2 p3 p4
  p4 p5 l2 offers p4,
  p5, p6
• s6 p2 p3 p4 p5 p6
• s7 p5 p3 p8 l3 s1 s7 d3 2
• l3 offers p7, p8
• c1 2 all other projects have capacity 1
• s7 applies to p3
• l1 becomes over-subscribed l1 rejects s3

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49
Execution of the algorithm

• Student preferences Lecturer preferences
• s1 p1 p7 l1 s7 s4 s1 s3 s2 s5
  s6 d1 3
• s2 p1 p2 p3 p4 p5 p6
  p3 p2 p1 l1 offers p1, p2, p3
• s3 p2 p1 p4
• s4 p2 l2 s3 s2 s6 s7 s5 d2 2
• s5 p1 p2 p3 p4
  p4 p5 l2 offers p4,
  p5, p6
• s6 p2 p3 p4 p5 p6
• s7 p5 p3 p8 l3 s1 s7 d3 2
• l3 offers p7, p8
• c1 2 all other projects have capacity 1
• s7 applies to p3
• l1 becomes over-subscribed l1 rejects s3
• p3 becomes full
• l1 becomes full p1 deleted from list of s3

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50
Execution of the algorithm

• Student preferences Lecturer preferences
• s1 p1 p7 l1 s7 s4 s1 s3 s2 s5
  s6 d1 3
• s2 p1 p2 p3 p4 p5 p6
  p3 p2 p1 l1 offers p1, p2, p3
• s3 p2 p1 p4
• s4 p2 l2 s3 s2 s6 s7 s5 d2 2
• s5 p1 p2 p3 p4 p4 p4 p5
  l2 offers p4, p5, p6
• s6 p2 p3 p4 p5 p6
• s7 p5 p3 p8 l3 s1 s7 d3 2
• l3 offers p7, p8
• c1 2 all other projects have capacity 1
• s3 applies to p4
• p4 becomes over-subscribed p4 rejects s2

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51
Execution of the algorithm

• Student preferences Lecturer preferences
• s1 p1 p7 l1 s7 s4 s1 s3 s2 s5
  s6 d1 3
• s2 p1 p2 p3 p4 p5 p6
  p3 p2 p1 l1 offers p1, p2, p3
• s3 p2 p1 p4
• s4 p2 l2 s3 s2 s6 s7 s5 d2 2
• s5 p1 p2 p3 p4 p4 p5
  l2 offers p4, p5, p6
• s6 p2 p3 p4 p5 p6
• s7 p5 p3 p8 l3 s1 s7 d3 2
• l3 offers p7, p8
• c1 2 all other projects have capacity 1
• s3 applies to p4
• p4 becomes over-subscribed p4 rejects s2
• p4 becomes full p4 deleted from list of s2
• l2 becomes full

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52
Execution of the algorithm

• Student preferences Lecturer preferences
• s1 p1 p7 l1 s7 s4 s1 s3 s2 s5
  s6 d1 3
• s2 p1 p2 p3 p4 p5 p6
  p3 p2 p1 l1 offers p1, p2, p3
• s3 p2 p1 p4
• s4 p2 l2 s3 s2 s6 s7 s5 d2 2
• s5 p1 p2 p3 p4 p4 p5 p5
  l2 offers p4, p5, p6
• s6 p2 p3 p4 p5 p6
• s7 p5 p3 p8 l3 s1 s7 d3 2
• l3 offers p7, p8
• c1 2 all other projects have capacity 1
• s2 applies to p5
• p5 becomes over-subscribed p5 rejects s6

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53
Execution of the algorithm

• Student preferences Lecturer preferences
• s1 p1 p7 l1 s7 s4 s1 s3 s2 s5
  s6 d1 3
• s2 p1 p2 p3 p4 p5 p6
  p3 p2 p1 l1 offers p1, p2, p3
• s3 p2 p1 p4
• s4 p2 l2 s3 s2 s6 s7 s5 d2 2
• s5 p1 p2 p3 p4 p4 p5 l2
  offers p4, p5, p6
• s6 p2 p3 p4 p5 p6
• s7 p5 p3 p8 l3 s1 s7 d3 2
• l3 offers p7, p8
• c1 2 all other projects have capacity 1
• s2 applies to p5
• p5 becomes over-subscribed p5 rejects s6
• p5 becomes full p5 deleted from list of s6
• l2 becomes full p6 deleted from list of s6

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54
Execution of the algorithm

• Student preferences Lecturer preferences
• s1 p1 p7 l1 s7 s4 s1 s3 s2 s5
  s6 d1 3
• s2 p1 p2 p3 p4 p5 p6
  p3 p2 p1 l1 offers p1, p2, p3
• s3 p2 p1 p4
• s4 p2 l2 s3 s2 s6 s7 s5 d2 2
• s5 p1 p2 p3 p4 p4 p5 l2
  offers p4, p5, p6
• s6 p2 p3 p4 p5 p6
• s7 p5 p3 p8 l3 s1 s7 d3 2
• l3 offers p7, p8
• c1 2 all other projects have capacity 1
• Algorithm terminates
• Matching is stable

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55
Pseudocode of algorithm (1)

• assign each student to be free
• assign each project and lecturer to be totally
  unsubscribed
• 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 /
• provisionally assign si to pj /
  and to lk /
• if (pj is over-subscribed)
• sr worst student assigned to pj
• break provisional assignment between sr
  and pj
• else if (lk is over-subscribed)
• sr worst student assigned to lk
• pt project assigned sr
• break provisional assignment between sr
  and pt

56
Pseudocode of algorithm (2)

• if (pj is full)
• sr worst student assigned to pj
• for (each successor st of sr on lks
  list)
• delete pj from sts list
• if (lk is full)
• sr worst student assigned to lk
• for (each successor st of sr on lks
  list)
• for (each project pu ? Pk )
• delete pu from sts list
• / while loop /

57
Theoretical results

• Algorithm produces student-optimal stable
  matching, given an instance of SPA
• Algorithm may be implemented to run in O(L) time,
  where L is total length of the students
  preference lists
• Second algorithm finds lecturer-optimal stable
  matching
• Same set of students are matched in all stable
  matchings
• Each lecturer obtains the same number of students
  in all stable matchings
• A project offered by an under-subscribed lecturer
  has the same number of students in all stable
  matchings

58
Open problems

• Extend to the case where lecturers have
  preferences over (student,project) pairs
• Ties in the preferences lists
• Lower bounds on projects
• Complexity of finding a greedy maximum matching