Semi-Matchings
for Bipartite Graphs
and Load Balancing
Nick Harvey, Richard Ladner,
Laszlo Lovasz, Tami Tamir
Talk Outline
 “Swiss
Bank” Problem
 Formal Definitions
 Optimal Semi-Matchings
 Algorithms
 Experiments
“Swiss Bank” Problem

5 bank tellers


each speaks a different language
10 bank customers

each speaks one or more languages

Assume servicing a customer takes
1 time unit

Problem: Assign each customer to a teller
Problem Model: Bipartite Graph
Customers
Tellers
German
Customer
speaks
German and
Italian
Italian
French
Romansh
English
Customer Assignment
Customers
Tellers
German
Italian
French
Romansh
English
Optimization Objectives
Minimize:
1. Flow time:

2.
Makespan:

3.
Total time customers wait
(or average time)
Maximum time a customer waits
Variance:

Load balance of tellers’ queue lengths
Customer Wait Time (Flow Time)
Wait Time
Customer
Queues
Tellers
1
German
1
Italian
4+3+2+1=10
French
2+1=3
Romansh
2+1=3
English
Total Wait Time:
1+1+10+3+3 = 18 units
Square of
difference
from mean
Teller Load and Variance
Customer
Queues
Tellers
Load Variance
German
1
(1-2)2=1
Italian
1
(1-2)2=1
French
4
(4-2)2=4
Romansh
2
(2-2)2=0
English
2
(2-2)2=0
Mean Load
10/5 = 2
Variance:
(1+1+4+0+0)/5 = 6/5
Optimal Assignment
Customers
Tellers
German
Italian
French
Romansh
English
Variance: 0
Total Wait Time: (2+1)*5 = 15 units
Talk Outline





“Swiss Bank” Problem
Formal Definitions
Optimal Semi-Matchings
Algorithms
Experiments
Formal Definitions

Let G = (U V, E) be a bipartite graph
U
V
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
Matchings


First studied by Philip Hall of
Cambridge University
“Marriage Theorem” characterizes the
existence of perfect matchings
Philip Hall
(1904-1982)
P. Hall, On representatives of subsets,
J. London Math. Soc. 10 (1935), 26-30.
Matchings

“Hungarian Algorithm” used to find
matchings of maximum cardinality
Harold W. Kuhn
Princeton University
H. W. Kuhn, The Hungarian method for the assignment
problem, Naval Res. Logist. Quart. 2:83-97, 1955.
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 semimatching, M
Max-Weight Semi-Matchings

Let w: E  R be an edge-weight function

Problem (Lawler ‘76): Find semi-matching
M maximizing
 w(e)
eM

Solvable by simple greedy algorithm
Eugene Lawler, Combinatorial Optimization: Networks
and Matroids. Holt, Rinehart & Winston, 1976.
Optimal Semi-Matching, M



Edges are unweighted
Let degM(v) denote number of Medges incident with vV
Define cost of M at a vertex vV:
c M (v ) 
deg M ( v )
 i  1  2  ...  deg
i 1
c
M
(v )
(v )

Define c(M) =

M is an optimal semi-matching if c(M)
is minimal
vV
M
Optimal Semi-Matchings

c(M) gives the total weighting time of the
“customers” in U
U
V
cM(v)
1
1
4+3+2+1=10
2+1=3
2+1=3
c(M) = 1+1+10+3+3 = 18
Talk Outline



“Swiss Bank” Problem
Formal Definitions
Optimal Semi-Matchings





Cost Reducing Paths
Optimality Criterion
Lp-norm
Algorithms
Experiments
Optimality Properties

Optimal Semi-Matchings have useful
load balancing properties:
1.
2.
3.

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
Alternating Paths


Let P = (v1, u1, …, uk-1, vk) be a path in G
If {vi, ui} M and {ui, vi+1} E \ M for all i,
then P is called an alternating path
U
V
v3
u2
u1
P is an
alternating
path
v2
v1
White edges
are in E \ M
Blue edges
are in M
Cost-Reducing Paths (CRPs)

Let P be an alternating path in G

If degM(vk) < degM(v1)-1 then P is called a
cost-reducing path
U
V
v3
degM(v3) = 1
u2
u1
P is not a
cost-reducing
path
v2
v1
degM(v1) = 2
Cost-Reducing Paths (CRPs)

Let P be an alternating path in G

If degM(vk) < degM(v1)-1 then P is called a
cost-reducing path
U
V
v2
degM(v2) = 1
v1
degM(v1) = 4
u1
P is a costreducing path
Improvement with CRPs



Let P = (v1, u1, …, uk-1, vk) be a CRP
Remove {vi, ui} from M for all i
Add {ui, vi+1} to M for all i
U
V
v2
degM(v2) = 2
1
v1
degM(v1) = 43
u1
P is a costreducing path
Improvement with CRPs



cM(v1) decreases by degM(v1)
cM(vk) increases by degM(vk)+1
Total decrease is (degM(v1)-degM(vk)-1) > 0
U
V
v2
cM(v2) = 3
1
v1
cM(v1) = 10
6
u1
c(M)
Decrease of 4
+ Increase of 2
= Decrease of 2
Talk Outline



“Swiss Bank” Problem
Formal Definitions
Optimal Semi-Matchings





Cost Reducing Paths
Optimality Criterion
Lp-norm
Algorithms
Experiments
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 semimatching – this was just shown
 If a semi-matching is not optimal then a
cost-reducing path exists – this requires
some proof
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
 = (M \ O)  (O \ M)
 Let
G be G restricted to edge-set 
Construction of G

Direct red edges VU and green edges UV
Suboptimal M
M\O
Optimal O
G

O\M
Properties of G
 Acyclicity
 G
contains no alternating red/green cycle
 Monotonicity
 If
there is an alternating red/green
path in G from v1 to v2 in V then
degO(v1)  degO(v2)
Both
properties hold by choice of O
Properties of G
G
O
degO(v)
2
• Acyclic
2
• Monotone
2
2
2
G yields CRP for M
G
M
degM(v)
3
There is a
cost-reducing
red/green path
for M
2
3
2
0
Existence of CRP Proof

Choose V-vertex v1 such that
degM(v1) > degO(v1)

Build red/green path in G until
we find V-vertex v2 with degM\O(v2) = 0
degM(v2) ≤ degO(v2) - 1
≤ degO(v1) - 1
< degM(v1) - 1

arrived at v2 on O\M edge
monotonicity
choice of v1
Path from v1 to v2 is a cost-reducing
path for M!
Talk Outline



“Swiss Bank” Problem
Formal Definitions
Optimal Semi-Matchings





Cost Reducing Paths
Optimality Criterion
Lp-norm
Algorithms
Experiments
Lp-norm of load vector
Let xi = degM(vi)
 The Lp-norm of the vector X=(x1,x2,…,xm) is

||X ||p = (i xip)1/p

||X ||1 is always |U |

For any p>1, ||X ||p is a measure of
the 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

Optimality of Lp-norm

Theorem: Let p>1. A semi-matching is
optimal iff the Lp-norm of its load
vector is optimal

Proof outline: Based on following claims
1. A cost-reducing path can reduce the Lpnorm of the load vector
Proof: Simple calculation
2. 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
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
0
1
2
1
Optimal L
Optimal semi-matching
Total cost: 6
Total cost: 5
Talk Outline




“Swiss Bank” Problem
Formal Definitions
Optimal Semi-Matchings
Algorithms




Network Flow Algorithms
Algorithm SM1
Algorithm SM2
Experiments
Network Flow Algorithms

Can reduce semi-matching problem
to known network-flow problems
1.
Assignment Problem
Requires O(n0.5 m . log(n)) time
(Gabow and Tarjan, 1989)
2.
Min-cost Max-flow Problem
Requires O(n .m . log2(n)) time
(Goldberg and Tarjan, 1987)
where n=num vertices and
m=num edges
Assignment Problem
G
Assignment
Problem
Min-cost Max-flow Problem
Source
U
V
Cost
Centers
Sink
Talk Outline




“Swiss Bank” Problem
Formal Definitions
Optimal Semi-Matchings
Algorithms




Network Flow Algorithms
Algorithm SM1
Algorithm SM2
Experiments
Algorithm SM1
 Simple
modification of “Hungarian
Algorithm” for Bipartite Matching
 Runtime: O(n .m) (n=|U |+|V | and m=|E |)
 Same
 Actual
as Hungarian Algorithm
performance is not good
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
Algorithm SM1 Example
U
V
1
1
Initially: no one is
assigned.
2
Step 1: assign
u1 to a least
loaded V-vertex
3
4
5
Algorithm SM1 Example
U
V
1
2
1
2
3
4
5
Step 2:
assign u2
Algorithm SM1 Example
U
V
1
2
3
1
Step 3:
assign u3
2
Can increase
the load on v2
3
or v1 or v3
4
v3 is the
least loaded
5
Algorithm SM1 Example
U
V
1
2
3
1
2
3
4
5
Talk Outline




“Swiss Bank” Problem
Formal Definitions
Optimal Semi-Matchings
Algorithms




Network Flow Algorithms
Algorithm SM1
Algorithm SM2
Experiments
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!
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
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.
Step 1: Greedy Example
U
V
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 |)
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 |)
Step 2: Find CRP Example
U
V
This vertex
has load 2.
No CRP yet.
This vertex
has load 1.
CRP has
been found!
This vertex
has the
highest load
(3 matched
neighbors)
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
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 |)



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?


Coin Towers Example
Tower
1
Tower
2
Tower
3
Tower
4
Tower
5
Total: 8 Moves
Tower
6
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


Tower K has maximum height C/K. Thus,
each coin moves at most C/K times after
passing Tower K



Because each move goes right
Because each move goes down
For arbitrary K, can prove that
each coins moves at most K+C/K times
Fix K=sqrt(C). Then maximum possible
moves is O(C . sqrt(C)) = O(C1.5)
Talk Outline





“Swiss Bank” Problem
Formal Definitions
Optimal Semi-Matchings
Algorithms
Experiments
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
Computing Optimal Semi-Matching (FewG Graph)
1000
Runtime (s)
100
SM1
10
SM2
CSA
LEDA
1
32768
65536
131072
0.1
0.01
Number of Vertices
262144
524288
Computing Optimal Semi-Matching (Grid Graph)
10
Runtime (s)
1
SM1
32768
65536
131072
262144
524288
SM2
CSA
LEDA
0.1
0.01
Number of Vertices
Computing Optimal Semi-Matching (Hilo Graph)
100
10
Runtime (s)
SM1
SM2
CSA
LEDA
1
32768
65536
131072
0.1
Number of Vertices
262144
524288
Computing Optimal Semi-Matching (Hexa Graph)
100
10
Runtime (s)
SM1
SM2
1
CSA
32768
65536
131072
0.1
0.01
Number of Vertices
262144
524288
LEDA
Computing Optimal Semi-Matching (ManyG Graph)
100
10
Runtime (s)
SM1
SM2
1
CSA
32768
65536
131072
0.1
0.01
Number of Vertices
262144
524288
LEDA
Computing Optimal Semi-Matching (Rope Graph)
100
10
Runtime (s)
SM1
SM2
CSA
LEDA
1
32768
65536
131072
0.1
Number of Vertices
262144
524288
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
Computing Maximum Matching (FewG Graph)
100
10
Runtime (s)
SM1
SM2
1
BFS
32768
65536
131072
0.1
0.01
Number of Vertices
262144
524288
LO
Computing Maximum Matching (Grid Graph)
10
SM1
Runtime (s)
1
32768
65536
131072
262144
524288
SM2
BFS
LO
0.1
0.01
Number of Vertices
Computing Maximum Matching (Hexa Graph)
100
Runtime (s)
10
SM1
SM2
1
BFS
32768
65536
131072
0.1
0.01
Number of Vertices
262144
524288
LO
Computing Maximum Matching (Hilo Graph)
10
Runtime (s)
1
32768
65536
131072
262144
524288
SM1
SM2
LO
BFS
0.1
0.01
Number of Vertices
Computing Maximum Matching (ManyG Graph)
100
Runtime (s)
10
SM1
SM2
1
LO
32768
65536
131072
0.1
0.01
Number of Vertices
262144
524288
BFS
Computing Maximum Matching (Rope Graph)
10
Runtime (s)
1
SM1
32768
65536
131072
262144
524288
SM2
LO
BFS
0.1
0.01
Number of Vertices
Computing Maximum Matching (Zipf Graph)
1000
Runtime (s)
100
SM1
10
SM2
LO
BFS
1
32768
65536
131072
0.1
0.01
Number of Vertices
262144
524288
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
Questions?
Algorithm SM1 Example
1
2
u
3
• Build an alternating tree
rooted at u. Edges (ui,vj)
are in E\M and edges (vj,ui)
are M.
1
2
3
4
5
v
• Select v, the least loaded
V-vertex in the tree.
•Re-assign matching edges
on path from u to v
Algorithm SM1 Example
U
V
1
2
3
4
1
Step 4: assign u4
2
Can increase the
load on v1 or v3
or v1 or v2
3
or v2
4
All have the
same load.
5
Algorithm SM1 Example
U
V
1
2
3
4
1
2
3
4
5
Assign u4 to v3
Algorithm SM1 Example
U
V
1
2
3
4
5
1
Step 5: assign u5
Step 6: assign u6
2
3
6
4
5
Algorithm SM1 Example
U
V
1
2
3
4
5
1
2
3
6
7
4
5
Step 7: assign u7
Can increase
the load on v3
or v1 or v2
or v1
or v2
v1 is the least
loaded
Algorithm SM1 Example
U
V
1
2
3
4
5
1
2
3
6
7
4
5