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Maximal Independent Sets of a Hypergraph

Maximal Independent Sets of a Hypergraph. IJCAI01. 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 IJCAI01

  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 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. … here’s some code

  14. So?

  15. You can reformulate a csp as a problem of finding a independent set of a hypergraph (this is not new) The independent set has to be of size n It is also maximal

  16. An Example X + Y + Z = 2 where X, Y and Z are in {0,1} Give me an independent set of size n We have n.m vertices A hyper edge for each nogood An m-clique for each variable’s domain

  17. Golomb Ruler Another Example • A ruler with N “ticks” • Distance between every pair of ticks is different • Ruler is of length L Independent Set encoding N.L vertices, corresponding to the L possible values for each of the N ticks N cliques of size L (binary nogoods) Hyper edges of arity 3

  18. Conclusion • There are maximality problems out there • e.g. determining properties of block design problems • experiments are proceeding • Don’t need maximality constraint for csp reformulation • but does it help propagation? • Experiments required • Reformulate a part of the problem, and link via channeling • Use maximality as learning? • For a csp with n variables, what is a maximal independent set of size less than n? What kind of nogood is this? • Do we have one way of dealing with solvable and over-constrained csp’s

  19. Easy questions only

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