Maximal Independent Sets of a Hypergraph

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Maximal Independent Sets of a Hypergraph
IJCAI01
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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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heres some code
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So?
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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
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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 variables domain
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Another Example
Golomb 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
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Conclusion

• 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

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