1 Algorithms for Student-Project Allocation David Manlove University of Glasgow Department of...

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

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

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 Capacities

Case Students

Lecturers

Projects

Lecturers

Case 1: formal definition

No explicit preferences; project capacities only

Set of students S={s1, s2, …, sn} Set of projects P={p1, p2, …, pm} Set of lecturers L={l1, 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

Definition of a matching

An assignment M is a subset of S×P 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:1. if si is assigned to pj in M then si finds pj acceptable

2. si is assigned to at most one project in M

3. pj is assigned at most cj students in M

Case 1: example

Set of students, projects and lecturers: S={s1, s2, s3, s4, s5}, P={p1, p2, p3, p4, p5}, L={l1, l2, l3}

Lecturers offer projects as follows: P1={p1, p2}, P2={p3}, P3={p4, p5}

Project capacities: c1= c2 = c4 = c5 = 1, c3 =2

Students find projects acceptable as follows: A1={p1, p3} A2={p1 , p4} A3={p2, p4} A4={p1, p2, p4} A5={p2, p4}

Corresponding bipartite graph

capacitys1

A matchingcapacity

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

A maximum matchingcapacity

Degree-constrained subgraph problem:A maximum matching may be found in time Owhere L is the number of edges (Gabow, 1983)

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 student’s

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.

Case 2: example

Set of students, projects and lecturers: S={s1, s2, s3}, P={p1, p2, p3}, L={l1}

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

Case 2: example

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

Case 2: example

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

Finding a greedy matching

Traditional method: transform to instance of assignment problem

Create weighted bipartite graph G=(V, E) with V=SP and E={(si, pj) : pjAi}

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 k=ranki(pj) Compute a maximum weight matching in G Complexity of algorithm: O(rn(L+nlog n))

(Fredman and Tarjan, 1987) Two problems:

Arithmetic operations involving edge weights O(r) Possible implementation difficulties

A direct algorithm

Combinatorial algorithm: O(min(n+R, Rn)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

Greedy matchings vs maximum matchings

Greedy matchings need not be maximum matchings

E.g. set of students, projects and lecturers: S={s1, s2, s3}, P={p1, p2, p3}, L={l1}

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

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

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

Greedy matchings vsrank-minimum matchings

E.g. set of students, projects and lecturers: S={s1, s2, s3,, s4, s5}, P={p1, p2, p3, p4, p5}, L={l1}

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

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:

1. if si is assigned to pj in M then si finds pj acceptable

2. si is assigned to at most one project in M

3. pj is assigned at most cj students in M

4. lk is assigned at most dk students in M

An optimal solution is a rank-minimum matching

Finding a rank-minimum matching (1)

Define a weighted network N as follows:

Vertices are V={s}SPL{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

The cost of a flow f is cost(f)={cost(e)f(e) : eE} 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(e+vlog v)log B/(m+n)) where v is number of vertices, e is number of edges and B is sum of edge capacities in N (Goldfarb and Jin, 1999)

Finding a rank-minimum matching (2)

Case 3: example Set of students, projects and lecturers:

S={s1, …, s5}, P={p1, …, p7}, L={l1, l2, l3}

Lecturers offer projects as follows: P1={p1, p2}, P2={p3, p4}, P3={p5, p6, p7}

Project capacities: p5 has capacity 2; all others have capacity 1

Lecturer capacities: d1=2, d2=1, d3=2

Student preference lists: s1 : p1 p3 p5 s2 : p1 p4 p6 s3 : p4 p1 p5 s4 : p1 p6 p7 s5 : p5 p3 p2

Example network N

Edge costs in greenEdge capacities in red

All these edgeshave capacity 1

Min cost-max flow f in N

Blue edges have flow >0- flow is 1 unless stated otherwise

cost(f)=10

Case 4:student 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

Definition of a blocking pair

(si, pj)M is a blocking pair of M if:

1. pj Ai

2. Either si is unmatched in M, or si prefers pj to M(si)3. Either

a) pj is under-subscribed and lk is under-subscribed

b) pj is under-subscribed and lk is full, and either siM(lk) or lk prefers si to the worst student in M(lk)

c) 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)

Example SPA instance

Student preferences Lecturer preferencess1 : p1 p7 l1 : s7 s4 s1 s3 s2 s5 s6 d1 = 3s2 : p1 p2 p3 p4 p5 p6 l1 offers p1, p2,

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

SPA instance: blocking pair

Student preferences Lecturer preferencess1 : p1 p7 l1 : s7 s4 s1 s3 s2 s5 s6 d1 = 3s2 : 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

(s1, p1) forms a blocking pair: s1 prefers p1 to M(s1)=p7

p1 is under-subscribed and l1 is under-subscribed

s3 : p2 p1 p4

s4 : p2 l2 : s3 s2 s6 s7 s5 d2 = 2

s6 : p2 p3 p4 p5 p6

s7 : p5 p3 p8 l3 : s1 s7 d3 = 2 p7 p8 l3 offers p7, p8

p5 is under-subscribed and s6 is assigned to a project offered by l2

Student preferences Lecturer preferencess1 : p1 p7 l1 : s7 s4 s1 s3 s2 s5 s6 d1 = 3s2 : 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

s6 : p2 p3 p4 p5 p6

s7 : p5 p3 p8 l3 : s1 s7 d3 = 2 p7 l3 offers p7, p8

p1 is under-subscribed and l1 is full and l1 prefers s1 to the worst student assigned to l1

s3 : p2 p1 p4

s4 : p2 l2 : s3 s2 s6 s7 s5 d2 = 2

s6 : p2 p3 p4 p5 p6

s7 : p5 p3 p8 l3 : s1 s7 d3 = 2 p7 l3 offers p7, p8

(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

SPA instance: stable matching

s3 : p2 p1 p4

s4 : p2 l2 : s3 s2 s6 s7 s5 d2 = 2

s6 : p2 p3 p4 p5 p6

s7 : p5 p3 p8 l3 : s1 s7 d3 = 2l3 offers p7, p8

The matching is stable

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)

Execution of the algorithm

Student preferences Lecturer preferencess1 : p1 p7 l1 : s7 s4 s1 s3 s2 s5 s6 d1 = 3s2 : 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

s6 : p2 p3 p4 p5 p6

Student preferences Lecturer preferencess1 : p1 p7 l1 : s7 s4 s1 s3 s2 s5 s6 d1 = 3s2 : 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

s6 : p2 p3 p4 p5 p6

s1 applies to p1

p1 remains under-subscribedl1 remains under-subscribed

s3 : p2 p1 p4

s4 : p2 l2 : s3 s2 s6 s7 s5 d2 = 2

s6 : p2 p3 p4 p5 p6

s2 applies to p1

p1 becomes full; p1 deleted from list of s5

l1 remains under-subscribed

s3 : p2 p1 p4

s4 : p2 l2 : s3 s2 s6 s7 s5 d2 = 2

s6 : p2 p3 p4 p5 p6

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

Student preferences Lecturer preferencess1 : p1 p7 l1 : s7 s4 s1 s3 s2 s5 s6 d1 = 3s2 : 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

s6 : p2 p3 p4 p5 p6

s4 applies to p2

p2 becomes over-subscribed; p2 rejects s3

s3 : p2 p1 p4

s4 : p2 l2 : s3 s2 s6 s7 s5 d2 = 2

s6 : p2 p3 p4 p5 p6

s4 applies to p2

l1 remains full

s3 : p2 p1 p4

s4 : p2 l2 : s3 s2 s6 s7 s5 d2 = 2

s6 : p2 p3 p4 p5 p6

s3 applies to p1

s3 : p2 p1 p4

s4 : p2 l2 : s3 s2 s6 s7 s5 d2 = 2

s6 : p2 p3 p4 p5 p6

s3 applies to p1

l1 remains full; p3 deleted from list of s2

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

s2 applies to p4

p4 becomes full; p4 deleted from list of s5 and s6

l2 remains under-subscribed

s3 : p2 p1 p4

s4 : p2 l2 : s3 s2 s6 s7 s5 d2 = 2

s6 : p2 p3 p4 p5 p6

s6 applies to p5

l2 becomes full

s3 : p2 p1 p4

s4 : p2 l2 : s3 s2 s6 s7 s5 d2 = 2

s6 : p2 p3 p4 p5 p6

s7 applies to p3

l1 becomes over-subscribed; l1 rejects s3

s3 : p2 p1 p4

s4 : p2 l2 : s3 s2 s6 s7 s5 d2 = 2

s6 : p2 p3 p4 p5 p6

s7 applies to p3

l1 becomes over-subscribed; l1 rejects s3

p3 becomes fulll1 becomes full; p1 deleted from list of s3

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

s3 applies to p4

s3 : p2 p1 p4

s4 : p2 l2 : s3 s2 s6 s7 s5 d2 = 2

s6 : p2 p3 p4 p5 p6

s3 applies to p4

l2 becomes full

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

s2 applies to p5

s3 : p2 p1 p4

s4 : p2 l2 : s3 s2 s6 s7 s5 d2 = 2

s6 : p2 p3 p4 p5 p6

s2 applies to p5

l2 becomes full; p6 deleted from list of s6

s3 : p2 p1 p4

s4 : p2 l2 : s3 s2 s6 s7 s5 d2 = 2

s6 : p2 p3 p4 p5 p6

Algorithm terminates

Matching is stable

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 si’s 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 ; }

Pseudocode of algorithm (2)

if (pj is full) { sr = worst student assigned to pj ; for (each successor st of sr on lk’s list) delete pj from st’s list ; } if (lk is full) { sr = worst student assigned to lk ; for (each successor st of sr on lk’s list) for (each project pu Pk ) delete pu from st’s list ; }} /* while loop */

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

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

1 Algorithms for Student-Project Allocation David Manlove University of Glasgow Department of...

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