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Semi-Matchings for Bipartite Graphs and Load Balancing

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Title: Semi-Matchings for Bipartite Graphs and Load Balancing


1
Semi-Matchingsfor Bipartite Graphsand Load
Balancing
  • Nick Harvey, Richard Ladner, Laszlo Lovasz, Tami
    Tamir

2
Talk Outline
  • Swiss Bank Problem
  • Formal Definitions
  • Optimal Semi-Matchings
  • Algorithms
  • Experiments

3
Swiss Bank Problem
  • 5 bank tellers
  • each speaks a different language
  • 10 bank customers
  • each speaks one or more languages
  • Assume servicing a customer takes1 time unit
  • Problem Assign each customer to a teller

4
Problem Model Bipartite Graph
Tellers
Customers
German
Customer speaks German and Italian
Italian
French
Romansh
English
5
Customer Assignment
Tellers
Customers
German
Italian
French
Romansh
English
6
Optimization Objectives
  • Minimize
  • Flow time
  • Total time customers wait(or average time)
  • Makespan
  • Maximum time a customer waits
  • Variance
  • Load balance of tellers queue lengths

7
Customer Wait Time (Flow Time)
Wait Time
1
1
432110
213
213
Total Wait Time 111033 18 units
8
Square of difference from mean
Teller Load and Variance
Load
Variance
(1-2)21
1
(1-2)21
1
(4-2)24
4
(2-2)20
2
(2-2)20
2
Variance (11400)/5 6/5
Mean Load 10/5 2
9
Optimal Assignment
Tellers
Customers
German
Italian
French
Romansh
English
Variance 0
Total Wait Time (21)5 15 units
10
Talk Outline
  • Swiss Bank Problem
  • Formal Definitions
  • Optimal Semi-Matchings
  • Algorithms
  • Experiments

11
Formal Definitions
  • Let G (U ?V, E) be a bipartite graph

U
V
12
Matchings
  • M ? E is a matching if each vertex is incident
    with at most one edge in M

U
V
The blue edges are a matching, M
13
Matchings
  • First studied by Philip Hall of Cambridge
    University
  • Marriage Theorem characterizes the existence of
    perfect matchings

P. Hall, On representatives of subsets, J. London
Math. Soc. 10 (1935), 26-30.
14
Matchings
  • Hungarian Algorithm used to find matchings of
    maximum cardinality

H. W. Kuhn, The Hungarian method for the
assignment problem, Naval Res. Logist. Quart.
283-97, 1955.
15
Semi-Matchings
  • M ? E is a semi-matching if each U-vertex is
    incident with exactly one edge in M

U
V
The blue edges are a semi-matching, M
16
Max-Weight Semi-Matchings
  • Let w E ? R be an edge-weight function
  • Problem (Lawler 76) Find semi-matching M
    maximizing
  • Solvable by simple greedy algorithm

Eugene Lawler, Combinatorial Optimization
Networks and Matroids. Holt, Rinehart Winston,
1976.
17
Optimal Semi-Matching, M
  • Edges are unweighted
  • Let degM(v) denote number of M-edges incident
    with v?V
  • Define cost of M at a vertex v?V
  • Define c(M)
  • M is an optimal semi-matching if c(M) is minimal

18
Optimal Semi-Matchings
  • c(M) gives the total weighting time of the
    customers in U

U
V
cM(v)
1
1
432110
213
213
c(M) 111033 18
19
Talk Outline
  • Swiss Bank Problem
  • Formal Definitions
  • Optimal Semi-Matchings
  • Cost Reducing Paths
  • Optimality Criterion
  • Lp-norm
  • Algorithms
  • Experiments

20
Optimality Properties
  • Optimal Semi-Matchings have useful load balancing
    properties
  • Minimize variance of degM(v)
  • Minimize max degM(v)
  • Minimize Lp-norm of degM(v)
  • Optimal Semi-Matchings contain a maximum matching
    as a subset
  • And a max matching is easy to find

21
Alternating Paths
  • Let P (v1, u1, , uk-1, vk) be a path in G
  • If vi, ui? M and ui, vi1? E \ M for all
    i,then P is called an alternating path

U
V
v3
u2
White edges are in E \ M
v2
u1
P is an alternating path
v1
Blue edges are in M
22
Cost-Reducing Paths (CRPs)
  • Let P be an alternating path in G
  • If degM(vk) lt degM(v1)-1 then P is called a
    cost-reducing path

U
V
v3
degM(v3) 1
u2
v2
u1
P is not a cost-reducing path
v1
degM(v1) 2
23
Cost-Reducing Paths (CRPs)
  • Let P be an alternating path in G
  • If degM(vk) lt degM(v1)-1 then P is called a
    cost-reducing path

U
V
v2
degM(v2) 1
u1
v1
degM(v1) 4
P is a cost-reducing path
24
Improvement with CRPs
  • Let P (v1, u1, , uk-1, vk) be a CRP
  • Remove vi, ui from M for all i
  • Add ui, vi1 to M for all i

U
V
v2
degM(v2) 1
degM(v2) 2
u1
v1
degM(v1) 4
degM(v1) 3
P is a cost-reducing path
25
Improvement with CRPs
  • cM(v1) decreases by degM(v1)
  • cM(vk) increases by degM(vk)1
  • Total decrease is (degM(v1)-degM(vk)-1) gt 0

U
V
v2
cM(v2) 1
cM(v2) 3
u1
v1
cM(v1) 10
cM(v1) 6
c(M) Decrease of 4 Increase of 2 Decrease of
2
26
Talk Outline
  • Swiss Bank Problem
  • Formal Definitions
  • Optimal Semi-Matchings
  • Cost Reducing Paths
  • Optimality Criterion
  • Lp-norm
  • Algorithms
  • Experiments

27
Optimality Criterion
  • Theorem A semi-matching is optimal if and only
    if no cost-reducing path exists
  • Proof
  • Any CRP can reduce the cost of a semi-matching
    this was just shown
  • If a semi-matching is not optimal then a
    cost-reducing path exists this requires some
    proof

28
Proof of Optimality Criterion
  • Let M be a suboptimal semi-matching
  • Let O be an optimal semi-matching with smallest
    symmetric difference ? with M
  • In ? color the edges of M red and the edges in O
    green
  • Let G? be G restricted to edge-set ?

? (M \ O) ? (O \ M)
29
Construction of G?
  • Direct red edges V?U and green edges U?V

Suboptimal M
Optimal O
G?
?
M \ O
O \ M
30
Properties of G?
  • Acyclicity
  • G? contains no alternating red/green cycle
  • Monotonicity
  • If there is an alternating red/greenpath in G?
    from v1 to v2 in V thendegO(v1) ? degO(v2)
  • Both properties hold by choice of O

31
Properties of G?
G?
O
  • Acyclic
  • Monotone

32
G? yields CRP for M
G?
M
There is acost-reducing red/green path for M
33
Existence of CRP Proof
  • Choose V-vertex v1 such thatdegM(v1) gt degO(v1)
  • Build red/green path in G? untilwe find V-vertex
    v2 with degM\O(v2) 0
  • Path from v1 to v2 is a cost-reducing path for M!

degM(v2) degO(v2) - 1 degO(v1) -
1 lt degM(v1) - 1
  • arrived at v2 on O\M edge
  • monotonicity
  • choice of v1

34
Talk Outline
  • Swiss Bank Problem
  • Formal Definitions
  • Optimal Semi-Matchings
  • Cost Reducing Paths
  • Optimality Criterion
  • Lp-norm
  • Algorithms
  • Experiments

35
Lp-norm of load vector
  • Let xi degM(vi)
  • The Lp-norm of the vector X(x1,x2,,xm) is
  • X 1 is always U
  • For any pgt1, X p is a measure ofthe balance
    of the load on V-vertices
  • X 2 is the sum of squares
  • X ? is the load of the most loaded V-vertex
  • X p is everything in between

X p (?i xip)1/p
36
Optimality of Lp-norm
  • Theorem Let pgt1. A semi-matching is optimal iff
    the Lp-norm of its load vector is optimal
  • Proof outline Based on following claims
  • A cost-reducing path can reduce the Lp-norm of
    the load vector
  • Proof Simple calculation
  • A semi-matching M has optimal Lp-norm iff no
    cost-reducing path relative to M exists
  • Proof Similar to the proof for optimal total
    cost

37
Optimality of L?-norm
  • Theorem An optimal semi-matching is optimal with
    respect to L? (load on most loaded teller)
  • Proof more complicated
  • The converse does not hold

xi
xi
2
2
1
0
1
2
Optimal L?
Optimal semi-matching
Total cost 5
Total cost 6
38
Talk Outline
  • Swiss Bank Problem
  • Formal Definitions
  • Optimal Semi-Matchings
  • Algorithms
  • Network Flow Algorithms
  • Algorithm SM1
  • Algorithm SM2
  • Experiments

39
Network Flow Algorithms
  • Can reduce semi-matching problem to known
    network-flow problems
  • Assignment ProblemRequires O(n0.5 m . log(n))
    time(Gabow and Tarjan, 1989)
  • Min-cost Max-flow ProblemRequires O(n .m .
    log2(n)) time(Goldberg and Tarjan, 1987)
  • where nnum vertices andmnum edges

40
Assignment Problem
Assignment Problem
G
41
Min-cost Max-flow Problem
Cost Centers
U
V
Source
Sink
42
Talk Outline
  • Swiss Bank Problem
  • Formal Definitions
  • Optimal Semi-Matchings
  • Algorithms
  • Network Flow Algorithms
  • Algorithm SM1
  • Algorithm SM2
  • Experiments

43
Algorithm SM1
  • Simple modification of Hungarian Algorithm for
    Bipartite Matching
  • Runtime O(n .m) (nU V and mE )
  • Same as Hungarian Algorithm
  • Actual performance is not good

44
Algorithm SM1 Pseudocode
  • Initially M is empty
  • For each u?U
  • Build tree T of alternating paths rooted at u
  • Let v be a V-vertex in T such that degM(v) is
    minimum
  • Switch matching and non-matching edges on path
    from v to u
  • Note u is matched and M increased by one

45
Algorithm SM1 Example
U
V
1
Initially no one is assigned.
1
Step 1 assign u1 to a least loaded V-vertex
2
3
4
5
46
Algorithm SM1 Example
U
V
1
Step 2 assign u2
1
2
2
3
4
5
47
Algorithm SM1 Example
U
V
1
Step 3 assign u3
1
2
3
Can increase the load on v2
2
or v1 or v3
3
v3 is the least loaded
4
5
48
Algorithm SM1 Example
U
V
1
1
2
3
2
3
4
5
49
Talk Outline
  • Swiss Bank Problem
  • Formal Definitions
  • Optimal Semi-Matchings
  • Algorithms
  • Network Flow Algorithms
  • Algorithm SM1
  • Algorithm SM2
  • Experiments

50
Algorithm SM2
  • General idea Find and remove cost-reducing paths
  • Runtime O(U 3/2 . E )
  • Worse bound than Algorithm SM1
  • Actual performance is very good!

51
Algorithm SM2 Pseudocode
  • Quickly find an initial semi-matching M
  • While M contains a cost-reducing path P
  • Improve M by switching edges along P
  • Stop M is optimal

52
Step 1 Initial Semi-Matching
  • Any semi-matching will work but a near-optimal
    one is better
  • Easy approach
  • Match each u?U with its least-loaded V-neighbor
  • Better approach
  • Sort vertices in U by increasing degree
  • Match each u?U with its least-loaded
    V-neighbor.In case of a tie, choose V-neighbor
    with least degree.

53
Step 1 Greedy Example
U
V
54
Step 2 Find CRP
  • Easy approach
  • For each v?V
  • Build tree T of alternating paths rooted at v
  • If T contains a cost-reducing path, return it
  • Return false
  • Runtime O(V . E )

55
Step 2 Find CRP
  • Better approach
  • Build forest F of alternating paths where each
    tree root is a least-loaded V-vertex that is not
    in F
  • If F contains a cost-reducing path, return it
  • Return false
  • Runtime O(E )

56
Step 2 Find CRP Example
U
V
57
Algorithm SM2 Analysis
  • Step 1 Find Greedy Matching O(E )
  • Step 2 Find CRP O(E )
  • Step 3 Eliminate CRP O(U V )
  • How many CRPs must be eliminated to achieve
    optimality?
  • Depends on cost of Greedy Assignment

58
Algorithm SM2 Num Iterations
  • Worst-possible Greedy Assignment has Total Cost
    U . (U 1)/2
  • Each iteration reduces Total Cost by at least 1
  • Therefore at most O(U 2) iterations
  • Total Runtime O(U 2 . E )
  • Can prove tighter bound O(U 3/2 . E )

59
Coin Towers Problem
  • Start Tower of coins C stories tall
  • Goal C towers of coins each 1 story tall
  • Coins can only move down and right
  • Minimum number of moves is obviously C-1
  • Problem What is maximum number of moves?

60
Coin Towers Example
Tower 1
Tower 2
Tower 3
Tower 4
Tower 5
Tower 6
Total 8 Moves
61
Coin Towers Analysis
  • Assume tower heights non-increasing from left to
    right
  • For any K, each coin moves at most K times before
    passing beyond Tower K
  • Because each move goes right
  • Tower K has maximum height C/K. Thus, each coin
    moves at most C/K times after passing Tower K
  • Because each move goes down
  • For arbitrary K, can prove thateach coins moves
    at most KC/K times
  • Fix Ksqrt(C). Then maximum possible moves is O(C
    . sqrt(C)) O(C1.5)

62
Talk Outline
  • Swiss Bank Problem
  • Formal Definitions
  • Optimal Semi-Matchings
  • Algorithms
  • Experiments

63
Semi-Matching Experiments
  • Compute Optimal Semi-Matchings
  • Compare SM1 SM2 to reduction to assignment
    problem
  • CSA Goldberg Kennedy, 1993
  • LEDA www.algorithmic-solutions.com
  • Use input graph generators from Cherkassky et
    al., 1998

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Maximum Matching Experiments
  • Compute Maximum Matchings
  • Compare SM1 SM2 to existing matching algorithms
  • BFS Breadth-First Search based alternating-path
    algorithm
  • LO Push-relabel algorithm with Lo heuristic
  • Both from Cherkassky et al., 1998
  • Use input graph generators from Cherkassky et
    al., 1998

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Conclusions
  • Optimal Semi-Matchings solve simple load
    balancing problems
  • Minimize maximum load and variance
  • Optimal Semi-Matchings contain Maximum Bipartite
    Matchings
  • Algorithm SM1 has an efficient theoretical bound
  • Algorithm SM2 is efficient in practice at
    computing Optimal Semi-Matchings and Maximum
    Matchings

79
Questions?
80
Algorithm SM1 Example
1
  • Build an alternating tree rooted at u. Edges
    (ui,vj) are in E\M and edges (vj,ui) are M.
  • Select v, the least loaded V-vertex in the tree.
  • Re-assign matching edges on path from u to v

1
2
u
3
2
v
3
4
5
81
Algorithm SM1 Example
U
V
1
Step 4 assign u4
1
2
Can increase the load on v1 or v3
3
2
4
or v1 or v2
or v2
3
All have the same load.
4
5
82
Algorithm SM1 Example
U
V
1
Assign u4 to v3
1
2
3
2
4
3
4
5
83
Algorithm SM1 Example
U
V
1
Step 5 assign u5 Step 6 assign u6
1
2
3
2
4
5
3
6
4
5
84
Algorithm SM1 Example
U
V
1
Step 7 assign u7
1
2
Can increase the load on v3
3
2
or v1 or v2
4
or v1
5
3
or v2
6
v1 is the least loaded
4
7
5
85
Algorithm SM1 Example
U
V
1
1
2
3
2
4
5
3
6
4
7
5
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