Kuhn Munkres algorithm Actors A definition There is a label on - - PowerPoint PPT Presentation

Actors

Kuhn Munkres algorithm

A definition

There is a label on every vertex and its value is L[x]. For any edge (x,y) we make sure that L[x] + L[y] ≥ w(x,y).

Another definition

We define the equivalent subgraph G'. G' includes all the vertices and all the tight edges. tight: w(x,y) = L[x] + L[y] Actually, we don't care about weight in the G'.

An important lemma

If the equivalant subgraph G' has a perfect match, then this match will be the maximum weighted match in G.

New goal

Our goal is to find the prefect match in the G'. Welcome Nayun Xu!

A simple description of algorithm procedure

• Step1. Set a label L[x] for every vertex x.
• Step2. Get the equivalent subgraph by definition.
• Step3. If every vertex has been matched , go to Step4. Find a vertex x not matched yet in the left

part, and try to find a match vertex y in the right part in the equivalent subgraph. If it is impossible to find a match vertex y for x, then we adjust the vertex label, and go back to Step2, else repeat Step3.

• Step4. The label now can lead to an equivalent subgraph with a perfect match. And the sum of all

the vertex label now is the value of max weight matching.

More details with an example

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More details with an example

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More details with an example

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More details with an example

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More details with an example

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More details with an example

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More details with an example

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More details with an example

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Key words in the algorithm

1.vertex label 2.equivalent subgraph 2.find match vertex (network flow or Hungarian algorithm) 3.adjust vertex label

Welcome Kevin Tan

A simple time bound and its proof

A brief introduction to dual problem

Actually ...

Thank you for listening!