Semi-Matchings for Bipartite Graphs and Load Balancing

1
Semi-Matchingsfor Bipartite Graphsand Load
Balancing

• Nick Harvey, Richard Ladner, Laszlo Lovasz, Tami
  Tamir

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Talk Outline

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

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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

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Problem Model Bipartite Graph
Tellers
Customers
German
Customer speaks German and Italian
Italian
French
Romansh
English
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Customer Assignment
Tellers
Customers
German
Italian
French
Romansh
English
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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

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Customer Wait Time (Flow Time)
Wait Time
1
1
432110
213
213
Total Wait Time 111033 18 units
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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
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Optimal Assignment
Tellers
Customers
German
Italian
French
Romansh
English
Variance 0
Total Wait Time (21)5 15 units
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Talk Outline

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

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Formal Definitions

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

U
V
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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
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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.
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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.
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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
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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.
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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

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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
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Talk Outline

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

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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

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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
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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
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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
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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
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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
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Talk Outline

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

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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

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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)
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Construction of G?

• Direct red edges V?U and green edges U?V

Suboptimal M
Optimal O
G?
?
M \ O
O \ M
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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

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Properties of G?
G?
O

• Acyclic

• Monotone

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G? yields CRP for M
G?
M
There is acost-reducing red/green path for M
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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

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Talk Outline

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

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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
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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

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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
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Talk Outline

• Swiss Bank Problem
• Formal Definitions
• Optimal Semi-Matchings
• Algorithms
• Network Flow Algorithms
• Algorithm SM1
• Algorithm SM2
• Experiments

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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

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Assignment Problem
Assignment Problem
G
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Min-cost Max-flow Problem
Cost Centers
U
V
Source
Sink
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Talk Outline

• Swiss Bank Problem
• Formal Definitions
• Optimal Semi-Matchings
• Algorithms
• Network Flow Algorithms
• Algorithm SM1
• Algorithm SM2
• Experiments

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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

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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

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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
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Algorithm SM1 Example
U
V
1
Step 2 assign u2
1
2
2
3
4
5
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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
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Algorithm SM1 Example
U
V
1
1
2
3
2
3
4
5
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Talk Outline

• Swiss Bank Problem
• Formal Definitions
• Optimal Semi-Matchings
• Algorithms
• Network Flow Algorithms
• Algorithm SM1
• Algorithm SM2
• Experiments

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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!

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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

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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.

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Step 1 Greedy Example
U
V
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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 )

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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 )

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Step 2 Find CRP Example
U
V
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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

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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 )

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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?

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Coin Towers Example
Tower 1
Tower 2
Tower 3
Tower 4
Tower 5
Tower 6
Total 8 Moves
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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)

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Talk Outline

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

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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

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Questions?
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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
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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
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Algorithm SM1 Example
U
V
1
Assign u4 to v3
1
2
3
2
4
3
4
5
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Algorithm SM1 Example
U
V
1
Step 5 assign u5 Step 6 assign u6
1
2
3
2
4
5
3
6
4
5
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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
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Algorithm SM1 Example
U
V
1
1
2
3
2
4
5
3
6
4
7
5
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