Student-Project Allocation with Preferences over Projects

1
Student-Project Allocation withPreferences over
Projects

• David Manlove
• Gregg OMalley
• University of GlasgowDepartment of Computing
  Science

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

• 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
• Lecturer preferences over students
• Project and lecturer capacities equal to 1
• University of York, Department of Computer
  Science
• Arbitrary project and lecturer capacities
• 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

4
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)
• Seek a stable matching as a solution
• Roth (1984)

5
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

6
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

7
Definition of a matching

• A matching M is a subset of SP such that
• if (si, pj)?M then pj ? Ai , i.e. si finds pj
  acceptable
• ? si ?S pj ?P (si, pj)?M ?1
• ? pj ?P si ?S (si, pj)?M ?cj ,
• ? lk ?L si ?S (si, pj)?M ? pj ? Pk ?dk
• 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 assigned student si , 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 (projects offered by) lk

8
Definition of a matching

• A matching M is a subset of SP such that
• if (si, pj)?M then pj ? Ai , i.e. si finds pj
  acceptable
• ? si ?S pj ?P (si, pj)?M ?1
• ? pj ?P si ?S (si, pj)?M ?cj ,
• ? lk ?L si ?S (si, pj)?M ? pj ? Pk ?dk
• 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 assigned student si , 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 (projects offered by) lk

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Definition of a matching

• A matching M is a subset of SP such that
• if (si, pj)?M then pj ? Ai , i.e. si finds pj
  acceptable
• ? si ?S pj ?P (si, pj)?M ?1
• ? pj ?P si ?S (si, pj)?M ?cj ,
• ? lk ?L si ?S (si, pj)?M ? pj ? Pk ?dk
• 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 assigned student si , 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 (projects offered by) lk

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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

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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

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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

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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

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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

15
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

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Sizes of stable matchings

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

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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)

18
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)

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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)

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Maximisation problem

• MAX-SPA-P denotes the problem of finding a
  maximum cardinality stable matching, given an
  instance of SPA-P
• There exists some ?gt1 such that the problem of
  approximating MAX-SPA-P within ? is NP-hard
• Result holds even if each project and lecturer
  has capacity 1, and all preference lists are of
  constant length
• Gap-preserving reduction from Minimum Maximal
  Matching (MMM)
• There exists some ?gt1 such that the problem of
  approximating MMM within ? is NP-hard
• Result holds even for subdivision graphs of cubic
  graphs
• Halldorsson, Irving, Iwama, DFM, Miyazaki, Morita
  and Scott, Approximability results for stable
  marriage problems with ties, Theoretical
  Computer Science, 2003

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Approximation algorithm
if (lk is over-subscribed)
/ lk prefers pj to pz / sr some
student in M(pz) M M \ (sr , pz)
delete pz from srs list
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
/ else / / while /

• 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 /
• pz lks worst non-empty project
• if (pj is full or (lk is full and pz pj ))
• delete pj from sis list
• else
• M M ? (si, pj)
• / si is provisionally assigned
• / to pj and lk /

22
Approximation algorithm
if (lk is over-subscribed)
/ lk prefers pj to pz / sr some
student in M(pz) M M \ (sr , pz)
delete pz from srs list
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
/ else / / while /

• 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 /
• pz lks worst non-empty project
• if (pj is full or (lk is full and pz pj ))
• delete pj from sis list
• else
• M M ? (si, pj)
• / si is provisionally assigned
• / to pj and lk /

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Approximation algorithm
if (lk is over-subscribed)
/ lk prefers pj to pz / sr some
student in M(pz) M M \ (sr , pz)
delete pz from srs list
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
/ else / / while /

• 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 /
• pz lks worst non-empty project
• if (pj is full or (lk is full and pz pj ))
• delete pj from sis list
• else
• M M ? (si, pj)
• / si is provisionally assigned
• / to pj and lk /

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Approximation algorithm
if (lk is over-subscribed)
/ lk prefers pj to pz / sr some
student in M(pz) M M \ (sr , pz)
delete pz from srs list
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
/ else / / while /

• 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 /
• pz lks worst non-empty project
• if (pj is full or (lk is full and pz pj ))
• delete pj from sis list
• else
• M M ? (si, pj)
• / si is provisionally assigned
• / to pj and lk /

25
Approximation algorithm
if (lk is over-subscribed)
/ lk prefers pj to pz / sr some
student in M(pz) M M \ (sr , pz)
delete pz from srs list
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
/ else / / while /

• 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 /
• pz lks worst non-empty project
• if (pj is full or (lk is full and pz pj ))
• delete pj from sis list
• else
• M M ? (si, pj)
• / si is provisionally assigned
• / to pj and lk /

26
Approximation algorithm
if (lk is over-subscribed)
/ lk prefers pj to pz / sr some
student in M(pz) M M \ (sr , pz)
delete pz from srs list
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
/ else / / while /

• 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 /
• pz lks worst non-empty project
• if (pj is full or (lk is full and pz pj ))
• delete pj from sis list
• else
• M M ? (si, pj)
• / si is provisionally assigned
• / to pj and lk /

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Theoretical results

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

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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