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Student-Project Allocation with
Preferences over Projects
David Manlove
University of GlasgowDepartment of Computing Science
Joint work with Gregg O’Malley
Supported by EPSRC grant GR/R84597/01,RSE / Scottish Exec Personal Research Fellowship
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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
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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
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Related work (2)
Lecturer preferences over students
1. 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)
2. 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
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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 O’Malley, “Student-Project Allocation with
Preferences over Projects”, Proc. ACID 2005, to appear Seek a stable matching as a solution
Roth (1984)
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Formal definition of SPA-P
Set of students S={s1, s2, …, sn} Set of projects P={p1, p2, …, pm} Set of lecturers L={l1, 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
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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
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An assignment M is a subset of S×P such that if (si, pj)M then si finds pj acceptable
Definition of a matching
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An assignment M is a subset of S×P 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
Definition of a matching
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An assignment M is a subset of S×P 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
Definition of a matching
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Definition of a matching
An assignment M is a subset of S×P 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
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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:
1. pj Ai
2. Either si is unmatched in M, or si prefers pj to M(si)
3. pj is under-subscribed and eithera) si M(lk) and lk prefers pj to M(si)
b) si M(lk) and lk is under-subscribed
c) 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
a) 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
b) 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
c) s4 M(l2) and l2 prefers p4 to his worst non-empty project
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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 preferencess1 : p1 p2 l1 : p1
s2 : p1 l2 : p2 (each project and lecturer has capacity 1)
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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 preferencess1 : 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 preferencess1 : p1 p2 l1 : p1
s2 : p1 l2 : p2 (each project and lecturer has capacity 1)
M1={(s1, p1)}
Student preferences Lecturer preferencess1 : 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
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
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Exact Maximal Matching
A matching M in a graph G is maximal if M{e} is not a matching for any edge eM
u1
u2
w1
w2
u3 w3
u1
u2
w1
w2
u3 w3
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?
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Overview of the reduction
ui
wj
must ensure that ui or wj matched
EXACT-MMinstance
assume n1
LH vertices
assume n2
RH vertices
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Overview of the reduction
ui
wj
must ensure that ui or wj matched
si : pj … y1…yn1-K qj : pj zj
ALL-SPA-Pinstance
EXACT-MMinstance
Student preferences
ti : z1…zn2
assume n1
LH vertices
assume n2
RH vertices
Lecturer preferences
(1 i n1) (1 j n2)
(1 i n2-K) rj : yj (1 j n1-K)
Each project and lecturer has capacity 1
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Overview of the reduction
ui
wj
must ensure that ui or wj matched
si : pj … y1…yn1-K qj : pj zj
ALL-SPA-Pinstance
EXACT-MMinstance
Student preferences
Each project and lecturer has capacity 1
ti : z1…zn2
assume n1
LH vertices
assume n2
RH vertices
Lecturer preferences
(1 i n1) (1 j n2)
(1 i n2-K) rj : yj (1 j n1-K)
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Approximation algorithm
M = M {(si, pj)} ; /* si is provisionally assigned to pj and to lk */ if (lk is full) { pz = lk’s worst non-empty project ; for (each successor pt of pz on lk’s list) for (each student sr such that srAt) delete pt from sr’s list ; }}
M = ;while (some student si is free and si has a nonempty list) { pj = first project on si’s list ; lk = lecturer who offers pj ; /* si applies to pj */ if (pj is full) delete pj from si’s list ; else if (lk is full) { pz = lk’s worst non-empty project ; if (pz = pj ) delete pj from si’s list ; else { sr = some student in M(pz); M = M \ {(sr , pz)} ; delete pz from sr’s list ;
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Approximation algorithm
M = M {(si, pj)} ; /* si is provisionally assigned to pj and to lk */ if (lk is full) { pz = lk’s worst non-empty project ; for (each successor pt of pz on lk’s list) for (each student sr such that srAt) delete pt from sr’s list ; }}
M = ;while (some student si is free and si has a nonempty list) { pj = first project on si’s list ; lk = lecturer who offers pj ; /* si applies to pj */ if (pj is full) delete pj from si’s list ; else if (lk is full) { pz = lk’s worst non-empty project ; if (pz = pj ) delete pj from si’s list ; else { sr = some student in M(pz); M = M \ {(sr , pz)} ; delete pz from sr’s list ;
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Approximation algorithm
M = M {(si, pj)} ; /* si is provisionally assigned to pj and to lk */ if (lk is full) { pz = lk’s worst non-empty project ; for (each successor pt of pz on lk’s list) for (each student sr such that srAt) delete pt from sr’s list ; }}
M = ;while (some student si is free and si has a nonempty list) { pj = first project on si’s list ; lk = lecturer who offers pj ; /* si applies to pj */ if (pj is full) delete pj from si’s list ; else if (lk is full) { pz = lk’s worst non-empty project ; if (pz = pj ) delete pj from si’s list ; else { sr = some student in M(pz); M = M \ {(sr , pz)} ; delete pz from sr’s list ;
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Approximation algorithm
M = M {(si, pj)} ; /* si is provisionally assigned to pj and to lk */ if (lk is full) { pz = lk’s worst non-empty project ; for (each successor pt of pz on lk’s list) for (each student sr such that srAt) delete pt from sr’s list ; }}
M = ;while (some student si is free and si has a nonempty list) { pj = first project on si’s list ; lk = lecturer who offers pj ; /* si applies to pj */ if (pj is full) delete pj from si’s list ; else if (lk is full) { pz = lk’s worst non-empty project ; if (pz = pj ) delete pj from si’s list ; else { sr = some student in M(pz); M = M \ {(sr , pz)} ; delete pz from sr’s list ;
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Approximation algorithm
M = ;while (some student si is free and si has a nonempty list) { pj = first project on si’s list ; lk = lecturer who offers pj ; /* si applies to pj */ if (pj is full) delete pj from si’s list ; else if (lk is full) { pz = lk’s worst non-empty project ; if (pz = pj ) delete pj from si’s list ; else { sr = some student in M(pz); M = M \ {(sr , pz)} ; delete pz from sr’s list ;
M = M {(si, pj)} ; /* si is provisionally assigned to pj and to lk */ if (lk is full) { pz = lk’s worst non-empty project ; for (each successor pt of pz on lk’s list) for (each student sr such that srAt) delete pt from sr’s list ; } } } }
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Approximation algorithm
M = ;while (some student si is free and si has a nonempty list) { pj = first project on si’s list ; lk = lecturer who offers pj ; /* si applies to pj */ if (pj is full) delete pj from si’s list ; else if (lk is full) { pz = lk’s worst non-empty project ; if (pz = pj ) delete pj from si’s list ; else { sr = some student in M(pz); M = M \ {(sr , pz)} ; delete pz from sr’s list ;
M = M {(si, pj)} ; /* si is provisionally assigned to pj and to lk */ if (lk is full) { pz = lk’s worst non-empty project ; for (each successor pt of pz on lk’s list) for (each student sr such that srAt) delete pt from sr’s list ; } } } }
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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
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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
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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
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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
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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 s3’s list
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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
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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 s2’s list
Example SPA-P instance
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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
Example SPA-P instance
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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
Example SPA-P instance
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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 s1’s list
Example SPA-P instance
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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
Example SPA-P instance
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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
Example SPA-P instance
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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 s5’s list
Example SPA-P instance
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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
Example SPA-P instance
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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 s5’s list
Example SPA-P instance
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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
Example SPA-P instance
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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 s6’s list
Example SPA-P instance
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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
Example SPA-P instance
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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 s6’s list
Example SPA-P instance
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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
Example SPA-P instance
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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
Example SPA-P instance
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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
Example SPA-P instance
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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
Example SPA-P instance
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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
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”
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A worst-case example
Student preferences Lecturer preferencess1 : 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
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A worst-case example
Student preferences Lecturer preferencess1 : 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)}, |M1|=n
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A worst-case example
Student preferences Lecturer preferencess1 : 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)}, |M2|=2n
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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
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