Weighted Bipartite Matching

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Weighted Bipartite Matching

• Lecture 4 Jan 18

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Weighted Bipartite Matching
Given a weighted bipartite graph, find a matching
with maximum total weight.
B
A
Not necessarily a maximum size matching.
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Todays Plan

• Three algorithms
• negative cycle algorithm
• primal dual algorithm
• augmenting path algorithm
• Applications

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First Algorithm
Find maximum weight perfect matching
How to know if a given matching M is optimal?
Idea Imagine there is a larger matching M
and consider the union of M and M
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First Algorithm

• Orient the edges (edges in M go up, others go
  down)
• edges in M having positive weights, otherwise
  negative weights

Then M is maximum if and only if there is no
negative cycles
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First Algorithm

• Key M is maximum ? no negative cycle

How to find efficiently?
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Complexity

• At most nW iterations
• A negative cycle in time by Floyd
  Warshall
• Total running time
• Can choose minimum mean cycle to avoid W

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Augmenting Path Algorithm

• Orient the edges (edges in M go up, others go
  down)
• edges in M having positive weights, otherwise
  negative weights

Find a shortest path M-augmenting path at each
step
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Augmenting Path Algorithm
Theorem each matching is an optimal k-matching.
Let the current matching be M Let the shortest
M-augmenting path be P Let N be a matching of
size M1
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Complexity

• At most n iterations
• A shortest path in time by Bellman
  Ford
• Total running time
• Can be speeded up by using Dijkstra O(m nlogn)
• Kuhn, Hungarian method

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Primal Dual Algorithm
What is an upper bound of maximum weighted
matching?
What is a generalization of minimum vertex cover?
weighted vertex cover
Minimum weighted vertex cover gt Maximum weighted
matching
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Primal Dual Algorithm
Minimum weighted vertex cover gt Maximum weighted
matching

• Primal Dual
• Maintain a weighted matching M
• Maintain a weighted vertex cover y
• Either increase M or decrease y until they are
  equal.

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Primal Dual Algorithm
Minimum weighted vertex cover gt Maximum weighted
matching

• Consider the subgraph formed by tight edges (the
  equality subgraph).
• If there is a tight perfect matching, done.
• Otherwise, there is a vertex cover.
• Decrease weighted cover to create more tight
  edges, and repeat.

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Complexity

• Augment at most n times, each O(m)
• Decrease weighted cover at most m times, each
  O(m)
• Total running time
• Egrevary, Hungarian method

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Quick Summary

• First algorithm, negative cycle, useful idea to
  consider symmetric difference
• Augmenting path algorithm, useful algorithmic
  technique
• Primal dual algorithm, a very general framework
• Why primal dual?
• How to come up with weighted vertex cover?
• Reduction from weighted case to unweighted case

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Faster Algorithms
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Bonus Question 2
(40) Find a maximum weighted k-matching which
can be extended to a perfect matching.
Relation to red-blue matching?
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Application of Bipartite Matching
Jerry
Darek
Tom
Isaac
Marking
Tutorials
Solutions
Newsgroup
Job Assignment Problem Each person is willing to
do a subset of jobs. Can you find an assignment
so that all jobs are taken care of?

• Ad-auction, Google, online matching
• Fingerprint matching

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Application of Bipartite Matching
With Halls theorem, now you can determine
exactly when a partial chessboard can be filled
with dominos.
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Application of Bipartite Matching

• Latin Square a nxn square, the goal is to fill
  the square
• with numbers from 1 to n
  so that
• Each row contains every number from 1 to n.
• Each column contains every number from 1 to n.

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Application of Bipartite Matching
Now suppose you are given a partial Latin Square.
Can you always extend it to a Latin Square?
With Halls theorem, you can prove that the
answer is yes.