1. Maximal Independent Sets of a Hypergraph IJCAI01

2. 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 Whats that then?

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

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

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

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

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

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

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

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

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

13. More Generally

14. … heres some code

18. So?

19. 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

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

21. Another ExampleGolomb Ruler 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

22. There are maximality problems out there e.g. determining properties of block design problems experiments are proceeding Dont 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 csps Conclusion

23. Easy questions only