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Maximal Independent Sets of a Hypergraph

Maximal Independent Sets of a Hypergraph. Alice & Patrick. What’s 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

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Maximal Independent Sets of a Hypergraph

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  1. Maximal Independent Sets of a Hypergraph Alice & Patrick

  2. What’s 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

  3. A Hypergraph Show Me! 1 9 2 3 4 5 7 8 6

  4. An Independent Set Show Me! 1 It aint maximal! 9 2 3 4 5 7 8 6 You could add vertex 3 or vertex 8!

  5. A Maximal Independent Set Show Me! 1 Now you’re talking! 9 2 3 4 5 7 8 6 There are 11 maximal independent sets of size 6

  6. The Largest Independent Set Show Me! 1 9 2 3 4 5 7 8 6 There is only one for this graph

  7. A Minimal Maximal Independent Set Show Me! 1 9 2 3 4 5 7 8 6 There are 3 minimal maximal independent set Honest!

  8. … and now for a constraint programming solution … in Choco

  9. CP/Choco 1 9 2 3 4 5 7 8 6 But what about maximality?

  10. Encoding Maximality CP/Choco 1 9 2 3 4 5 7 8 6 An example, vertex 2 That is, we state when a variable MUST be selected and when it MUST NOT be selected

  11. CP/Choco 1 9 2 3 4 5 7 8 6 Example, vertices 1,2, and 3

  12. More Generally

  13. So?

  14. Conclusion? It’s fun • 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?

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