Maximal Independent Sets of a Hypergraph

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Maximal Independent Sets of a Hypergraph
Alice Patrick
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Whats that then?

• A hypergraph G (V,E)
• V is a set of vertices
• E is a set of hyperedges
• an edge with 2 or more vertices

• An independent set S
• assume vertices(e) is set of vertices in
  hyperedge e

• Maximal independent set S
• there is no independent set S that subsumes S

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Show Me!
A Hypergraph
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Show Me!
An Independent Set
You could add vertex 3 or vertex 8!
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Show Me!
A Maximal Independent Set
There are 11 maximal independent sets of size 6
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Show Me!
The Largest Independent Set
There is only one for this graph
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Show Me!
A Minimal Maximal Independent Set
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and now for a constraint programming solution
in Choco
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CP/Choco
But what about maximality?
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CP/Choco
Encoding Maximality
An example, vertex 2
That is, we state when a variable MUST be
selected and when it MUST NOT be selected
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CP/Choco
Example, vertices 1,2, and 3
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More Generally
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So?
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Conclusion?

• Is this any good?
• What is state of the art?
• Are there benchmark problems?
• Is it new?
• In CP, I think so
• Can we apply the encoding to other maximality
  problems?
• I guess so. Care to suggest some?
• Is it fun?